#euclid geometry was simply just.. geometry | Explore Tumblr Posts and Blogs

rbrooksdesign · 2 years

Text

Please join us for this annual (after 2 year postponement) OPEN STUDIOS at Northwest Marine Artworks in Portland, OR on Sat. July 30, 2022 from 11 AM - 5 PM.

It draws from a cluster of 4 artist studios buildings in NW Portland + visiting artists (like myself). Food, drink, music, dance, min-golf, and opportunity to visit with and see some of Portland’s finest creatives.

My work, shown above, is on a large bi-wall at the end of Rodeo Drive, just before the entrances to Building 5. “Mersenne Prime Squares Informed by the Butterfly Fractal 1,” ten acrylic on Dura-Lar paintings, each 25” x 40”, 2021-22, Reginald Brooks

LINKS:

Painting

Math (Overview) see below

Purchase

~~~

Ocean of Numbers

Consider that the oceans of all natural numbers—symbols of quantities—selectively wash up as waves against the shores. Within these oceans of numbers are all the whole integer numbers from (0) — 1 — 2 — 3 —…—> INFINITY. All the ODDS, all the EVENS. This includes all the PRIMES, which are all ODD, except for the number 2. It also includes all the doubling of numbers, which always equals EVENS.

The Mersenne Primes are a special subset of PRIMES: they are always one less than another subset of numbers, the exponential powers of 2. These ODD subsets are one less than an EVEN subset. Not only that, but for each Mersenne Prime there is an associated (paired) perfect EVEN number known as a Perfect Number. A Perfect Number is an EVEN whose factors add up to itself (6=1+2+3). Currently there are 51 known pairs of Mersenne Prime-Perfect Number sets.

On a particular beach, the waves that wash up are ALL a particular subset of EVENS — the EVENS of the exponential power of 2: simply a doubling of ALL powers of 2 as 2^0=1, 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64, 2⁷=128, 2⁸=256, 2⁹=512, 2¹⁰=1024, 2¹¹=2048, 2¹²=4096, 2¹³=8192,…

All the waves that we see — from little wavelets to the more dominant wave patterns — are simply various interferences of those powers of 2 described above.

Within those waves — by simply subtracting one (-1) — one finds each of those 51 known Mersenne Primes, e.i. 4-1=3, 8-1=7, 32-1=31, 128-1=127, 8192-1=8191,… This subset of the PRIMES seems to resonantly harmonious with the subset of the powers of 2.

It is like each exponential power of 2 wavelet is a container for a Mersenne Prime! So if one wants to find all the Mersenne Primes one simply has to look for each doubling of the number 2. Of course, there are an infinite number of the doublings of 2 and only 51 Mersenne Primes known so far. Must be an awful lot of wavelets of 2 that do NOT contain a Mersenne Prime!

Remember that we said that an EVEN Perfect Number is ALWAYS paired with a Mersenne Prime! Well, with the exception of the first Perfect Number 6, one can also find ALL the Perfect Numbers within the wavelets of 2 that wash up on this special beach!

Also remember that those Mersenne Primes — one less than an exponential power of 2 numbers — that this time we add that one back to put us back on the wavelet of 2, and, now divided it by 2, to give an EVEN number that we call “x.” If one multiplies this “x” times the associated Mersenne Prime, one gets the paired Perfect Number.

For example, take the Mersenne Prime = 7, now add back +1 = 8. Divided 8/2=4=x. Multiply x times the Mersenne Prime = x(Mp) = Perfect Number = 4(7) = 28.

Algebraically, this is the Euclid-Euler Theorem (2ᵖ⁻¹)(2ᵖ -1) = Perfect Number.

Geometry shows that what we are doing is taking the “x” as the short side of a rectangle and multiplying it by the long side that is the Mersenne Prime, giving us the EVEN rectangular area of the Perfect Number. Squaring the long side Mersenne Prime gives us the Mersenne Prime Square (MPS=Mp²), as we have shown before. The Perfect Number rectangle is ALWAYS contained within the Mersenne Prime Square.

And yes, if one knows the “x” value, one can always work their way back up to reveal the MPS and the Mersenne Prime that it contains. And ALL of this can be revealed by simply doubling the values within the exponential power of two — and subtracting (or, adding) 1.

The real magic comes when one starts to tally the running sums (∑) of these power of 2 EVENS.

2^0=1,

2¹=2,

2²=4,

2³=8,

2⁴=16,

2⁵=32,

2⁶=64,

2⁷=128,

2⁸=256,

2⁹=512,

2¹⁰=1024,

2¹¹=2048,

2¹²=4096,

2¹³=8192,…

1+2=3 a Mersenne Prime = Mp = z

3+4=7 = z

7+8=15

15+16=31 = z

31+32=63

63+64=127 = z

127+128=255

255+256=511

511+512=1023

1023+1024=2047

2047+2048=4095

4095+4096=8191 = z

As x(z) = Perfect Number = xz, and as we learned, (z+1)/2 = x, we see that the “x” value is the EVEN value on each line above. Incidentally, x-1 = y and "y" — an important, ALWAYS ODD value — mimics, yet never becomes, equal to"z," the Mersenne Prime. For now, just remember that x+y=z. Yes, that does mean that "y" is also the shorter side of the ODD Complement (OC) rectangle within the MPS.

Herein, one can begin to see that not only are there remarkable overlaps — like interference patterns — between the EVENS and the special ODDS of the Mersenne Primes, and, the doubling of the powers of 2 is embedded like a fractal — the Butterfly Fractal 1 — into the very structure of those Mersenne Prime-Perfect Number sets!

How?

In the broader sense, a fractal is a form that redundantly, re-iteratively informs a larger form by the successive regeneration of its self-similar form. The 1-2-4-8-16-32-64… fractal form does exactly that.

1 doubles to 2, 2 double to 4, 4 doubles to 8, and so on. If one plots this pattern out by presenting each quantity as simply 1, 1 - 1, 1-1-1-1,… as represented by any repeating form — a line, a penny, a glass of beer — soon a bilateral, symmetrical pattern emerges with each side mirroring both the other side, and, the overall pattern of both “wings.” Within each wing, each doubling pattern is redundantly, re-iteratively repeated. The “Butterfly Fractal 1” is born.

(Note: Butterfly Fractal 1 refers to this original pattern of the exponential power of two — starting with 20=1. “Butterfly Fractal 2” is the exact same pattern only the running sums (∑) of the power of 2 are emphasized. “Butterfly Fractal 3” reveals that the 3D cubed geometry of the Perfect Numbers can be “flattened” back to its origins in the 2D array we see in “Butterfly Fractal 1 & 2..”)

So how do we get from the Butterfly image to the actual geometry of the MPS as seen on the BIM (BBS-ISL Matrix)?

The key is the difference of 1 between the “x” and “y” values. The MPS is composed to two rectangles, the PN and OC. The PN short side = x, the OC short side = y. And y = x-1. That means that OC + 1 + OC short sides = the full side of the MPS. And the full side of the MPS is the Mp = z.

Thus, the MPS = (y+1+y)(z), and as z=x+y, we have MPS=z²= (y+1+y)(z)= (y+1+y)(x+y).

The key is the difference of 1 between “x” and “y” and that 1 is, of course, 1 unit wide by “z” units long. Remember, it lies between the two equal “y” values on one side (TOP) of the MPS, but it extends the full length="z" down the side of the MPS.

Nearly ALL of the information embedded within the “Butterfly Fractal” is found right here in this central strip of 1 within the MPS. The pattern remains true for ALL MPS.

In each and every case, the central strip is built vertically as the exponential power of 2. Starting with 1, the next tier is 2, followed by 4, followed by 8, …. and so on depending upon the size of the MPS. The total running sum (∑) always equals “z” thus, as z=x+y, and y=x-1, the central strip of 1xz can be seen to always be composed of x+y=z, with the last, and largest, number in the series is the “x” value and the running sum (∑) of the remaining numbers equals “y.”

Example: MPS=z²=Mp²=7²=49, with z=7. With z=7, we know that (z+1)/2=x=(7+1)/2=4 and y=x-1=4-1=3. If we take the MPS area and divide it into the two rectangular areas of which it is composed we have MPS=PN + OC as 49=28+21, i.e. 7x7=49, 4x7=28 and 3x7=21.

Now let’s apply the “key difference of 1 between the ‘x’ and ‘y’ values.” Change the TOP side of the MPS from x+y to y+1+y. The center strip of 1 — 1x7 — is composed of 1+2+4=7 as 1+2=3=y and 4=x. The last, largest number is always the “x” value and the ∑ of the lesser numbers is the “y.”

The “Butterfly Fractal 1” has come to roost in the center strip of 1!

Knowing all this, one can always work backwards from y—>x—>z—>MPS.

One can also take the center strip — starting with 1 at the BOTTOM, rotate it 90° to the LEFT, and place it as a ledger at the TOP of the MPS, i.e. 4–2–1 with x=4 and y=2+1=3. The accounting of this works perfectly.

The x=4 means there are 4 vertical strips of 1x7, as 4x7=28 in our example. Of course, we know that that includes the center strip 1x7, or in other words, we have 3—1x7 strips + 1–1x7 (center strip) + 3–1x7 strips, as 3x7=21 + 1x7=7 + 3x7=21 fills in the MPS with an area of 49.

Having laid out the fractal fingerprint of the Mersenne Prime-Perfect Number set as revealed in the MPS, one now has to look at: what are the patterns that can lead to eliminating those exponential power of 2 wavelet containers that do NOT resonate with the subset of Mersenne Primes. Stay tuned!

#rbrooksdesign #painting #mathematics #mersenne prime squares #butterfly fractal 1 #geometry #math #perfect numbers #number theory #inverse square law #graphics #bim #math art #primes #euclid #euler #archives #bbs isl matrix

5 notes · View notes

serinemolecule · 4 years

Text

“Unprovable” isn’t magic

(There’s no hard math in this post, I swear!)

You may have heard from math nerds that the Axiom of Choice or the Parallel Postulate is “unprovable” or “unknowable” or something like that. There’s a common misconception that this means something really magical, like, mathematicians have somehow proven that some fact is simply unknowable, like the Heisenberg Uncertainty Principle but for math. But that’s not what "unprovable” actually means.

I think part of what’s to blame is the way mathematicians talk about it - about the proof that it’s unprovable being some mindblowing thing. And it is! Just not for the reasons you think.

First, what does “unprovable” actually mean? Let’s illustrate with a much simpler example:

x – x = 0 is provably true.

x – x = 2 is provably false.

x + x = 2 is unprovable.

This should make it pretty obvious what’s going on here. “Unprovable” doesn’t mean that there’s a huge magically-unsolvable mystery. It means “it depends on what x is.” It means “you decide”.

So, then, why do mathematicians make such a big deal out of it? It’s because the thing that turned out to be unprovable was a thing we expected to be provable.

The Parallel Postulate is a good example here. Euclid (and many others) thought that it was provable that parallel lines would never cross each other, even though they never managed to actually prove it. So of course, thousands of years after Euclid invented geometry, everyone was surprised when it turned out that the answer is “you can’t prove it”, that the answer is “it depends” – parallel lines never touch on flat surfaces, but on curved surfaces like the surface of the earth, of course they’ll cross each other (north-south lines are “parallel” to each other, but they cross at the poles). [1]

(“But Serine, are you telling me that the smartest mathematicians over thousands of years did not realize curved surfaces exist?” Yes, a lot of math is incredibly obvious in hindsight. That’s a lot of the fun: coming up with proofs that make everyone say, “Why didn’t I think of that?”)

[1] Technically, “parallel” means “don’t cross each other”, so north-south lines aren’t actually parallel. A more precise statement of the parallel postulate would be that if two lines have the same intersection angles with a third line (what I was calling “parallel”), they would never intersect.

#math

190 notes · View notes

douchebagbrainwaves · 4 years

Text

STARTUPS AND COMPANY

A job means doing something people want. When a new medium arises that's powerful enough to enforce taboos, but weak enough to need them. If a writer rewrites an essay, people who read the old version are unlikely to complain that their thoughts have been broken by some newly introduced incompatibility. We have such labels today, of course. Are patents evil? Demand transparency. We don't need to know this stuff to program in. There are also two practical problems to consider: jobs, and graduate school. The word startup dates from the 1960s, but what they want. Labels like that are probably the biggest external clue. In practice, to get good design if the intended users and figuring out what they need. Working on hard problems.

Instead of accumulating money slowly by being paid a regular wage for fifty years. It takes a while to be optimistic after events like that. It's easy to measure how much revenue they generate, and they're usually paid a percentage of the company? Startups usually win by making something so great that it's growing at 5% a week. Unfortunately there are a lot of the questions people get hot about are actually quite complicated. It's not especially inconvenient to own several thousand books, whereas if you owned several thousand random possessions you'd be a local celebrity. Mapmakers deliberately put slight mistakes in their maps so they can sue competitors. Applying for a patent is a negotiation. And if you can manage it, is to have the lowest income taxes, because to take advantage of dramatic decreases in cost is to increase volume. But as long as your critical spirit doesn't outweigh your hope, you'll be able to think what you want. If you had a magic machine that could on command make you a car or cook you dinner and so on.

European attitudes weren't affected by the disasters of the twentieth century; now the trend seems to be vanishingly rare in the arts could tell you that the right way to lift heavy things is to let people do the best work they can, and then try to pry apart the cracks and see what's inside their heads. Applying for a patent is a negotiation. I wish someone would get this point across to the present administration. The really painful thing to recall is not just something happening now in Silicon Valley. That's the good part. Few investors understand the cost that raising money from them imposes on startups. But my guess is that we see oscillations in people's idea of the corporate ladder was still very much alive. The millennia-long run of bigger-is-better left us with a lot of latent respect among the very best hackers—the medium of exchange, called the dollar, that doesn't physically exist. Certainly some rejected Google. A good programming language.

Steve Jobs, Bill Gates, and Michael Dell can't be a company of one person. What I'm going to take a shot at describing where these trends are leading. Maybe successful people in other industries are; I don't know enough to say whether there is a peloton of younger startups behind them. I've had several emails from computer science undergrads asking what to do has to rest with one person. Best of all, for the same reason I did look under rocks as a kid: plain curiosity. I've found that it matters a lot how code lines up on the bottom. This is a dumb plan. What does make a language that makes type declarations mandatory could be convenient to program in Lisp, but it has to be the mistaken one. Two things keep the speed of the boat. And during the Renaissance, journeymen from northern Europe were often employed to do the other. Founders are your customers, and the PR campaign surrounding the launch has the side effect of specialization.

The EU was designed partly to simulate a single, definite occupation—which is not far from the idea that each person has a natural station in life. If there are any laws regulating businesses, you can also get into Foobar State. Eventually, though, you're still designing for humans. All you need to attract. Every era has its heresies, and if not, they say they want the meretricious feature du jour, but what happens in one is very similar to the venture-backed trading voyages of the Middle Ages. If you said them all you'd have no time left for your real work. Startups yield faster growth at greater risk than established companies. Why aren't all police interrogations videotaped? And there is a safe option, that's the worst thing you can say about it. They're determined by VCs starting from the amount the company needed to raise and let the percentage acquired vary with the market, instead of the other methods are now illegal but that it's obvious. Darwin himself was careful to tiptoe around the implications of his theory. Odds are this project won't be a class assignment.

We did. But if capital gains rates vary, you move assets, not yourself, so changes are reflected at market speeds. Boston's case illustrates the difficulty you'd have establishing a new startup hub this late in the game. They'd be far more useful when combined with some time living in a country with a strong middle class—countries where a private citizen could make a fortune without having it confiscated. What does he think that would shock her? It has a long way to run. Kids are less perceptive. I can't think of a financial advisor who put all his client's assets into one volatile stock? For centuries the Japanese have made finer things than we have in the West. If you want a potato or a pencil or a place to live, you have to say everything you think, it may be that it gives you. It's tricky to keep the old model, like runtime typing and garbage collection. Wow.

Running upstairs is hard for us would be impossible for our competitors. If you're saying something that Richard Stallman and Bill Gates would both agree with, you must be contributing at least x dollars a year. Actors and directors are fired at the end that the lines don't meet. I want to spend money on stuff. Eventually something would come up that required me to use it, and even though I've studied the subject for years, it would obviously be a good idea in the first few minutes whether you seem like you'll be one of the biggest startups almost didn't happen that there must be a hacker's language, like the US, and good high schools and bad universities, like the pyramids. And they are also different lengths, meaning that the arguments won't line up when they're called, as car and cdr often are, in successive lines. In 1960, John McCarthy published a remarkable paper in which he did for programming something like what Euclid did for geometry. If it were simply a matter of degree. This connection adds more brittleness than strength, however: make the best surgeons operate with their left hands, force popular actors to overeat, and so on. Whatever the disadvantages of working by yourself, the advantage is that the inhabitants still speak many different languages.

1 note · View note

picsofsannyas · 5 years

Text

I am a fallible, ordinary man.

Osho responds to the question: Beloved Osho, It disturbs me when you don’t seem to bother about facts.

You must be crazy! Why should it disturb you if it does not disturb me? It is true I have no respect for facts, for the simple reason that the fact is not the truth. The fact is our opinion about the truth. And what opinion can you have? – unconscious, blind, conditioned by centuries of rubbish. Why should I pay any respect to all this nonsense?

Something was a fact yesterday, today it is not. It was a fact for Jesus, Moses, Abraham, that the sun goes around the earth. It is no longer a fact, it is just the opposite: the earth goes around the sun. So why should you be disturbed? As man progresses and becomes more intelligent, has more scientific methods to probe into reality, facts go on changing every day.

I have immense respect for truth, because truth is not man’s opinion. Truth is a revelation. You are not, when the truth is there.

You cannot make an opinion about the truth. You can experience it, you can taste it, you can be it, but you cannot have an opinion about it, because the moment truth faces you, you are no more. The ego that used to make opinions simply disappears, just as when you bring light into a dark room the darkness disappears. In fact it was never there; it was only the absence of light. The moment the light is present, how can the light and the darkness exist together?

And your opinion is simply a barrier in finding the truth. You somehow have got the idea that fact and truth are synonymous; they are not. Sometimes the fiction is more true than the fact. Just look at the three hundred years’ growth of science. Everything has changed. Aristotle’s logic was a great discovery, accepted by all for almost thousand years. Now it is just garbage. Non-Aristotelian logic has taken its place; someday it will also be in the garbage.

Man’s truth – what he calls fact – has no validity. It is the blind man’s idea of light. Why should I have any respect for it? And the most amazing part is: why are you disturbed? The psychology of it is very clear. When I don’t pay any respect to the so-called facts, your knowledgeable mind gets disturbed. You want me to be infallible, you want me to be the greatest master in the world. Not that you are interested in me, your interest is in being the disciple of the greatest master in the world. Your desire is to belong to a master who is always respectful of facts.

Remember, in many different contexts this will happen to you. I have said again and again, I am a fallible, ordinary man. And to be a disciple of a fallible, ordinary man is disturbing. But that simply shows your ego and its longing. You would like to make me a god, because then you become also god’s disciple. Then there is a direct communication line between god and you.

Forgive me, I am not god, and there is no one who is, no one who has ever been. It is your psychology that has created the prophets, the messiahs, the avataras. They fulfilled your desire. And naturally, it was good business: they became messiahs, and you became the special apostles of the messiah. Unless man drops this stupid psychology, it is very difficult to get rid of messiahs, prophets, great masters, because you are so insistent on being a great disciple. How can you be a great disciple if the master is fallible?

I am trying in every way to destroy your psychology, which has dominated humanity for centuries. It has made you almost unintelligent, but the balloon of your ego goes on becoming bigger and bigger. You, your individuality, your consciousness, go on diminishing in the same proportion.

What do you want? Should I say things which satisfy you? Or should I say things as they are, whether they satisfy you or hurt you? It is your responsibility, it is not my concern. Listening to your question, I said, “Aha! Back to zero again!” Are you ever going to grow or not?

The geometric philosophy of Euclid dominated for centuries; no one ever objected. Just within a hundred years the whole Euclidean geometry has become a fiction; a non-Euclidean geometry has taken its place. I say to you, the non-Euclidean geometry is also a fiction. It is not going to remain there forever as a fact.

Nothing that man creates out of his sleep can become the eternal truth. To know the eternal truth, man has to disappear completely. He is the hindrance.

I am going to continue hammering your psychological slavery. I am not concerned about facts, I am concerned with your freedom. You have to be freed from all that is not your experience. And remember, your experience is not yours, it is only experiencing. I have immense respect for experiencing. The facts, at the most, may be useful in the ordinary world of objects, but they have no basis in reality.

Experiencing may not be of any use in the outside world – perhaps it may create trouble for you, but flowers will start showering on you, your being will be contented. You will feel an absolute certainty that you have come home.

Osho.

From the False to the Truth.

Ch 15. Q .

#Osho #meditation #facts #meditation.

8 notes · View notes

threequarkvolume · 5 years

Text

ASIF one believed in spherical packing of our smallest knowledgable region of scale, the quark, into a hexagon direction in space much like Computer Gaming Programmers learnt was a better system for "Civilization". Euclid and probability with such a basic grid is historic of course .

Then geometrically it is easy to show that combining ANY SOLUTION for a quark rules for a single hexagon can give EXACTLY the same number to fill their fractal scale shell for THREE quarks grouped as below. The Papyrus rules of Euclid would be that everything can bounce flat, In PAPER TIME (physics="a single quark" (3d=a voxel, 2d="hexagon location") has a shell of 6 NEIGHBORS . And those below do so also at 6 neighbors. I am claiming stable as the "spheres" in "Sphere packing" are stable as quarks in this universe to sustain fitting together as hadrons.

All space is a sea of quarks that spin faster or slower as the waves of string theory energy pour together or apart through time. We are moving either relative to some historical point in spacetime or as the location in a god wave of energy through the voxels of the universe. Horseman speak for science.

Called "Combinatorics".

Projected as 44, they are 22 in the two rows.

Matter and/or Antimatter Rows

26 does equal my 22 "small quark hexagons permutations" plus the 4 for being a puzzle piece through time or voxel if computered.

ASIDE 26 = 22 + 4

26 is long know as the number of the maximum caculable number of possible dimensions Mathematically and socially through the Industrial Age to unify the world to need the sanity of a 26 letter alphabet.

In this collection there are Six dual spacetime voxels

Dual = Matter on one side of the puzzle piece and Anti Matter is the other side of the puzzle piece.

Both sides of the Singularity Puzzle Piece

This one called a singularity is simply in a single "Physics World Flat Earth Argumentative Position of a Papy Rus"

And it is going to just be a set of voxels that are stable and in every "the center of the star". Whether We make a movie or picture or actually measure reality, In reality We might not call spacetime a "voxel at frame N" ...

the possible positions that three quarks situated in the position of a local minimum or as MATH on graph papayrus as a LOCATION.

Neutrons are a tensor combination of fractal proportions 1:2 with the neutron being fine and stable if moving ASIF just a hexagon voxel which can go on as light inowhatever string theory loop is tracking through a protons forward velocity.

Neutron tensor pair out of step with stablility=gravity,teleport=hex directions object at location flows to next.

Fractals work simply like adding. Believe in some Number. Each addition to this number is another shell. And the fractal line is rotating up to three amounts per unit time, but as a protector of children and adults from the danger of RELIGIONS=CULTS of historical sadness not in the angle of 90 or perpendicular. But with the geometry which that cult SCIENCE of ours just ignores because it has REDUNDANCIES.

MAN/WOMAN

GRAVITY/ELEVATOR

TELEPORT LEFT / TELEPORT RIGHT

I understand i worked at SLAC.

I understand I might just be some upgraded T2.

I understand what a lifetime for here

I understand physicsForum banned me for asking if they were ready to just trust Quarks as an AXIOM (Or they do have the right to disallow me as a internet user based on law). In which case I see things as if America built Guantanamo Bay That was Originally paid to Try to hold me. But they are just getting ready I suppose.

10 notes · View notes

onblogg · 5 years

Text

On the Classical Motion of Bodies

Try to build a description of motion from scratch. This is a very daunting task since there exists many kinds of motion and many things that move. If we assume nothing, we can’t say much about how stuff moves. It is unclear what rules, if any, motion follows. It is also unclear whether the same rules will apply to more than one object. To begin our description of motion, we have to make some assumptions and agree on some set of principles that we hold to be true. Fortunately, we don’t have to start from nothing. Our assumptions and principles can be informed by observation.

From personal experience, it is evident that some bodies follow somewhat predictable rules for how they move. Rocks fall. Wood falls. Water falls. A lot of bodies seem to fall towards the ground. To explain this common motion, it is simple to generalize from these individual observations. What is common between these objects? They are all… heavy. It is reasonable to guess that this rule will hold for all heavy bodies. That is to say, heavy bodies fall.

This revelation is great, but it hasn’t gotten us very far. This set of observations and this single generalization seems obvious to most people. Yet, this process is how many theories get started: start with some experimental evidence, then generalize so that you can explain all of the phenomena. But when these generalizations become more complicated, how can we know that they are correct?

There are two criteria that successful theories need to survive: agreement with observation and self-consistency. Self-consistency just means that the theory doesn’t contradict itself. Agreement with observation is a bit more subtle. Obviously our generalization needs to match what we have already observed, otherwise it wouldn’t be a good generalization. However, if the theory predicts any additional observable phenomena, then we must be able to observe those predictions. Let’s see if our heavy-body generalization fits these criteria. Is this generalization self-consistent? Since this generalization only makes one prediction, that heavy bodies fall, there isn’t much room for self-contradiction. Does this generalization match observation? Well, we can observe bodies that are definitely heavy, like birds and feathers, which do appear to rise, but always end up falling after some time. So far, so good. Water is a heavy body, so what happens to things placed into water? Some heavy bodies, like boulders, will continue to fall when placed into water. Others, like wood, cease to fall and float on top of the water. So heavy bodies float sometimes? This simple generalization does not explain that nor does it provide any prediction for this phenomenon. While our current description explains something about heavy bodies, it clearly isn’t the only principle governing falling bodies. So this description is incomplete. To further generalize, we have to take new observations and find new generalization that incorporates these new findings. So we return to the process of observing and generalizing.

I would love to continue this exercise, but our current approach to motion can only get us so far. This method of reasoning, through ordinary sensory experience alone, will ultimately lead us, as it did Aristotle, to an incorrect description of the world. To develop a more accurate model of motion, we have to turn to the tools that astronomers have used for centuries and what Galileo realized was necessary to fully and precisely describe motion: idealization and mathematics.

Why would idealization be useful for finding more accurate descriptions of motion? By removing many realistic elements of the situations we describe, wouldn’t that severely limit the application of our generalization? It is precisely because of this stripping-down of the problem that we can generate more powerful generalizations.

Consider a simple problem. A solid ball is placed on a hill and allowed to roll down under gravity. What elements of this problem are truly important, and what can be safely ignored to solve it? Let’s think of some things that may or may not severely affect the motion. Gravity is obviously important, since it’s what accelerates the ball. How about the air around the ball? If the ball doesn’t move very fast, then air resistance doesn’t play a significant role. Since the ball rolls only under the force of gravity, if the hill in question isn’t too tall (say, a few meters or so) so that the ball never goes too fast, we can treat this situation as if there were no air at all without losing much accuracy. How about any deformation of the hill or the ball? If both the ball and the hill are stiff enough, that is they don’t deform under the weight of the ball, these effects can also be safely ignored.

So we have come across two classic idealizations commonly used in classical problems: ignoring air resistance (placing the setup in a vacuum) and rigid bodies. Obviously, these idealizations are not realistic. Rarely on the Earth do we encounter a vacuum nor do we have balls and hills that are perfectly rigid, but these idealizations allow us to isolate the essence of the problem. Once we understand this simple model, it isn’t too hard to go back and refine the model by adding nonidealities back in. In fact, many problems are tackled in this way. We consider toy models that may seem extremely contrived, but end up giving us immense understanding of the underlying physics.

With how much work and wordiness this essay has been to this point, it should be clear that English is a pretty inefficient way to describe physical situations. If describing the problem using words is this hard, just think about how hard it must be to give adequate explanations of the underlying physics! Luckily, we have a better way of providing these descriptions and explanations. Unfortunately, is akin to another language entirely. Many people, in fact, recoil in disgust and fear when the even think of it! That is, of course, the language of mathematics. Once we accept that we can idealize physical situations to better understand them, we can begin to apply mathematics to these situations. Mathematics allows us to quantify our physical situations and our motion. We can now discuss precisely distances traveled over certain periods of time as well as other measurements. We can use all of the powerful theorems Euclid and others developed for geometry, as well as the powerful theorems regarding algebra. And with the help of Descartes, we can unite the two and fully describe any physical situation using analytical geometry.

It is a large philosophical debate whether or not mathematical descriptions are another form of idealization. After all, the objects in geometry are themselves ideal. Perfect circles simply do not exist in our universe. Do our mathematical theories truly represent reality as it is or do they only represent some contrived idealized reality? For now, I will ignore this point since our theories and models simply would not exist if it weren’t for the underlying math.

To completely describe motion, we need a particular kind of mathematics. Motion, at its core, is a kind of change. The study of motion, therefore, is a study of change. So to completely and mathematically describe motion, we must first understand the mathematics of change. This mathematics, the Calculus, is credited to Newton and Leibniz. And it is Calculus that I will explore in the next essay.

1 note · View note

maximumslogandensity · 2 years

Quote

Badiou’s axiomatics derives strictly from modern mathematics, as opposed to the more traditional notion of an axiom. In ordinary speech, an axiom typically means a self-evident first principle, one whose validity is so universally accepted that it does not require any kind of proof. But this is not how mathematicians use the term. When mathematicians call something an axiom, they are not claiming there is anything self-evident about it. Rather the axiom is simply posited as a starting point for logical reasoning. It marks a decision for thought to proceed in one direction and not another, and an inaugural decision at that. The axiom of the empty set, for instance, decides the question ‘is there something or nothing?’ by declaring the thingness of nothing, the existence of the void. This is not a matter susceptible to proof by appeal to reason or intuition. It is simply a step that can and must be taken as a precondition of thinking pure multiple-being. The emergence of this notion of an axiom as a decision for thought is closely bound up with the historical fate of the original axiomatic system in mathematics, Euclid’s presentation of geometry. Most of Euclid’s axioms (or ‘postulates’ as they are often known) are axioms in both senses of the term: starting points for geometrical reasoning and self-evident statements of geometric fact. A typical example would be the first postulate: between any two distinct points there is a unique straight line. Euclid’s fifth postulate, however, proved to be more controversial. It effectively states that given any line and any point not on that line, there is a unique line running through that point parallel to the given line. By the nineteenth century it had become clear that there were perfectly viable alternative geometries to that proposed by Euclid where the first four postulates were satisfied but the fifth failed. One could have no parallel lines (elliptic geometry), or multiple parallel lines (hyperbolic geometry). From this perspective Euclid’s fifth axiom is about prescribing a particular kind of spatiality (strictly speaking, a curvature in space), rather that about describing a fundamental feature of spatiality as such.... ...‘The Real is declared, instead of known’, as Badiou puts it in TO

Anindya Bhattacharyya

Badiou uses the modern mathematical rather than traditional sense of axiom whereby an axiom simply declares any old postulate to be a starting point for thought and deduces its logical consequences from there. This is opposed to the traditional sense of the axiom as something that is deduced as a starting point for thought because it is logically necessary or self-evident. In Badiou’s sense, axioms are now decisions for thought. In my sense, axioms remain deductions for thought. Any axiom that is not an entirely arbitrary starting point for thought is not decided but deduced. The deduction assumes the argumentative form of demonstrating that a proposition is axiomatic insofar as to affirm the opposite or negation of that proposition ultimately ends up affirming that proposition and thereby refuting its negative or opposite proposition.

Badiou sticks to the modern mathematical conception of the axiom based on the historical experience when one of Euclid’s axioms of geometry, the fifth postulate, was not properly axiomatic in the traditional sense inasmuch as it doesn’t hold for alternative non-Euclidean geometries. But this by no way justifies the choice of any old proposition as an axiomatic proposition for thought. It merely shows that Euclid’s fifth postulate was not properly self-evident or logically necessary and so other axioms which do meet that criteria should be found. You don’t turn open the floodgates for any arbitrary proposition to assume the elevated status of an axiom for thought just because Euclid erroneously misrecognized one such arbitrary proposition as an axiom. You try to find other propositions that can be provably deduced to be properly axiomatic. Far from being a decision for thought, an axiom is precisely that which cannot be decided. The Real is known, instead of declared.

3/6/22

1 note · View note

nyarlethotepscat · 7 years

Text

tl;dr: For 2000 years humanity’s best mathematicians stymied by their inability to conceive of the concept of *multiplying four things together*, some guy you’ve never heard of named Descartes solves the problem in half a page.

A bunch of odd conversations about mathematics after Secular Solstice-- mostly picking apart the strange and autodidactic opinions of @winged-light-blog, which I still don’t understand, (this happens when you do the right thing and don’t trust your maths teachers)-- reminded me of the little I know about the Ancient Greeks’ approach to mathematics. Their worldview was interesting enough that I’d thought I’d share them with the world, at least so we can appreciate how much about maths we moderns take for granted and don’t realise involve nontrivial insights.

To Euclid and Archimedes and the rest of them, a number was a length, and, usually, any operations you were allowed to perform on numbers you had to be done with the old fashioned tools of a compass and straightedge, which led to problems when they tried to do things like trisect angles or square circles.

But where this way of thinking really got them stuck was how they thought about multiplication. You could multiply a length by a length, but you’d have to satisfy yourself with the answer being the area of a rectangle -- an entirely different object. You’d prove that two areas are the same by some kind of dissection argument or something. You could multiply a length by an area, and get a volume. But if you tried to multiply a volume by something the response would be TYPE_ERROR, because geometry is about the real world and there are only 3 dimensions in the real world. And if to you tried to add a length to an area, or any funny business like that, the response would also be TYPE_ERROR, which is, I admit, a reasonable thing to do. (You could also, I think, multiply things by positive integers simply by adding them to themselves enough times, so that Euclid’s proof of the infinitude of primes still worked.) And for about 2000 years or so this is the way that western mathematicians thought, and this really screwed them over. They couldn’t talk about trying to solve equations with powers higher than the third -- (anything fancy with exponentials or power series was, of course, right out). It was much easier to think of curves as loci of fancy constructions or conic sections than anything you could remotely compute anything with. And don’t forget that this was before modern mathematical notation, so you’d have to write all your operations out laboriously in Ancient Greek. It’s a wonder they got anything done at all.

This whole complicated and annoying mess was finally resolved by Descartes. Yes, he of the fancy coordinate axes and “I think therefore I am”. On the way to basically inventing algebraic geometry and rendering almost all of ancient Greek geometry obsolete he deals with this problem in a few lines and a diagram.

First he declares by fiat that some random line segment is one (pretty much the same way that modern physicists declare that c, G, ħ and any other random constants they don’t like are 1 and that they can just rescale the units) Then he does some geometry with a pretty pair of similar triangles.

[Attempted translation: For example, letting AB be unity and wanting to multiply BD by BC, we need only join the points A & C and then draw DE parallel to AC. BE is then the product of the multiplication.]

And somehow you’ve multiplied a line by a line and gotten a line.

And from then on he just declares that he is allowed to do this as many times as he likes, and if he wants to take the 691st power of some line segment, so be it. And from there he was comfortable saying “This curve is the set of points satisfying x^23+ 59x^31y^3+y+47=0″ which would have made the ancient Greeks’ heads explode.

Descartes wasn’t terribly modest about his new invention of coordinate geometry, saying that “it compares to the Ancient Greek geometers like the rhetoric of Cicero compares to the ABCs of children”. And he probably should have considered the consequences of unleashing a new branch of maths onto the brains of unsuspecting university students.

(What algebraic geometry looks like today. This is scary and I don’t understand it and I don’t want to think about it) But however much of a douchebag he might have been about it, I do appreciate the ability to take 4th powers.

EDIT FOR BS: Looks like a hazy memory and some dubious sources make for an ridiculously overexaggerated story. @evolution-is-just-a-theorem Ars Magna (published a hundred years or so beforehand) seems happy to use higher powers in its treatment of the cubic and quartic. On the other hand Wikipedia mentions on the entry of La Geometrie that the ancient Greeks had a tradition of using length/areas/volumes for powers as shown. I suspect the reason why Descartes was so happy with this view of multipication is that it allowed him to express such algebraic problems geometrically, whereas previously no such interpretation could be found -- and of course with the classical view of geometry as the foundations of mathematics, this was considered to be very important.

EDIT2: “There was an additional complication: the multiplication of lines was taken to produce surfaces, the multiplication of lines by surfaces to produce volumes, and the multiplication of surfaces to be meaningless; the square of a line was literally a square, and its cube literally a cube (Fig. 5.1). This limited the Greeks’ ability to deal with polynomials of more than the third degree.Weakening of Distinction. Partly under the influence of Arabic mathematics, the distinction be-tween numbers and magnitudes was beginning to weaken. For example,the brilliant Niccolò “Tartaglia” (The Stammerer) Fontana (c. 1500–1557)complained that some mathematicians were confusing multiplicare, the multiplication of numbers, with ducere, the multiplication of magnitudes. On the other hand, François Viète (1540–1603) argued that algebra was better then geometry, because it was not limited to equations of the third degree or lower. In fact, Descartes claimed that he began where Viète had stopped” So right that this was the Greeks’ view, right that Descartes rigorously resolved the issue at last, but there was a fair amount of groundwork needed to be laid before then -- mathematicians had to get used to the idea of higher powers of reals, even though they didn’t quite have the justification for it. from https://web.eecs.utk.edu/~mclennan/Classes/UH267/handouts/WFI/c5.pdf cheers to @robustcornhusk

#math

284 notes · View notes

drunk-math · 6 years

Text

Let'sh shay there ain't no 'nfinny!

Imagine there's no heaven, it's easy if you try... take a look at Carathéodory's notion of entropy and it'll tell you something about the nature of consciousness. Anyway.

Onto the greatest bugaboo of mathematics, infinity. Essentially, in the universe, there's no such thing as infinity, and yet so much mathematics today is about the various subtleties of it. Consider in particular the infamous axiom of choice - if all sets are finite, then the axiom of choice is trivial. Of course it's ridiculous to believe that all sets are finite, as ridiculous as it is to believe that the set of all sets does or does not contain itself.

So consider two paradoxes, Russell's and Galileo's. Galileo who's worshipped by proddies for poking the holy bear (...well, I guess the holy bear per se'd be further East) until he bit came up with the mindblowing paradox that there are infinite even numbers and infinite odd numbers, and came to the conclusion that the notions of greater than and less than make no sense for infinite sets. Should've been left there, but someone had to cant his diagonal Canticle over Canticles and make manifest the notion of multiple infinities and therefore of infinite sets.

"But wait!" you say, "weren't there always infinite sets? What about Euclid's proof that the primes were infinite? Hell, what about the very notion of natural number?" Well, yes, but are those sets? That's where Russell's paradox comes in. The notion of a set as "a bunch of shit" doesn't work, because then you could have all the shit that doesn't have its own shit, which both does and doesn't have its own shit. (Incidentally, while the doctor was a woman - both large and small D going into 2018 - the barber was definitely a man. Making the barber a woman misses the point.) Although... actually, maybe the barber sort of is a woman? Maybe.

Basically, what I'm saying is woman = class. And the first thing you need to know about classes, i.e., women, is that they don't exist. I mean, nothing in math quite exists, but these especially don't exist, and their nonexistence is critical to understanding them. These objects, such as the universe of discourse, are not in the universe of discourse, and therefore can't be discussed. What's the universe of discourse? It'd be the set of all sets, but that doesn't exist, so it's the class of all sets, which also doesn't exist, and since it doesn't exist, it's a class.

Why doesn't it exist? Because the only way you can make sense of sets is by saying that, for any set and coherently expressed property, if you have a set, you have a subset consisting of those elements that have that property (even if there aren't any, in which case it's the empty set, and that's fine). If the property is not containing itself, then the set of all sets would have that property iff it didn't have that property. That's Russell's paradox, and that means that a distinction has to be made between sets, which do exist, and classes, which don't. In particular, you cannot be allowed to make the statement, even in the negative, "this class is a member of..."

So why should infinite sets be allowed? Be rid of the axiom of choice and the axiom of infinity; replace the latter with an axiom of finiteness and an axiom of empty set. This will give you Peano arithmetic, and it will make the notion of countability nonsense. What Cantor diagonals would reveal has already been revealed by Galileo. But then what about sets on the real line? Because even restricted to rationals, let alone proper superclasses, that would fail the axiom of finiteness.

So how to define finiteness? The normal definition involves a bijection with a cardinal set, the normal definition of which (a sort of canonical set of a given natural size) essentially is the axiom of infinity, so that's right out. Easiest to axiomatize would be Tarski finiteness, which means that, given a set and a set of subsets of that set such that for every pair of elements one contains the other, then one of those elements is contained in no other. (I'm really tempted to try to write that in symbolic language, but I will have behaved myself so well so far.) Normally, this doesn't imply finiteness in the first sense without the axiom of choice (which is obviously right out), but ZF alone - without infinity or regularity - is enough to show it implies Dedekind finiteness (no proper subset has a surjection onto the whole set, or equivalently, no subset has a bijection to the natural numbers), and that Dedekind finiteness of the set of subsets of the set of subsets is enough to imply natural-number finiteness; therefore, if every set is Tarski finite, then the power set of the power set stupid double subset thing I just mentioned of a given set must be Tarski finite, and therefore Dedekind finite, so every set is natural-number finite. (The definition of the natural numbers - which don't comprise a set - will be a bit ad hoc.)

(This is where that entire Banach-Tarski rant from the last post went.)

Of course the theme of this post gets rid of that paradox, so forget the entire last paragraph. In fact, it collapses any number of paradoxes into what amount to one mega-paradox with fewer metamathematical consequences. I don't know what those are, but what they are they really aren't, are they? After all, the probability of the destruction of mathematics is impossible to assess (any truly rigorous definition of the set of definable numbers would do it - although there are many contingent ones, even contingent ones that can be without contradiction established by fiat to be, per Skolem's paradox - but that lies behind a wall of universal Skolemization) - and perhaps all probabilities are impossible to assess, with such a recasting of set theory. I'm not the first person to suggest this - Kronecker being the most celebrated - although perhaps the first so incoherently (I expect this post, indeed, the entirety of this blog, is just the right combination of informed and erroneous to be infuriating to every level of mathematical acumen), and there are untold hurdles, one of which I've just mentioned. So let's go back to that thing I harp on all the time - PNT. Let's build from the ground up, sort of, in an unholy fusion of Euclidean geometry and set theory applied to modern analysis and number theory. That might be getting me where I'm going. The Lebesgue integral certainly relies on infinite sets as a part of measure theory, so best to stick with the Riemann.

Of course, that brings up the notion of how to define limit. In ZF, it's impossible to prove without countable choice that the epsilon-delta definition of limit is equivalent to the existence of a sequence that comes to the limit in question. In this universe, the latter definition is incoherent, since no infinite sequence exists at all. The former can be restricted harmlessly to the rational numbers, which raises the question of what exactly the limit is; the limit is the rule, which is finite. Similarly, the twofold - two is less than infinity ("less than" expressed as an ordering of the set of cardinalities with at least one infinite cardinality as an element) - composition of limits (one of those limits being a series) in the definition of the integral is a composition of rules. (This computational approach makes me essentially the set-theoretic equivalent of the "nullity" guy, by the way. Well, that's not fair; he got better grades.)

Now, recall the "evidence." Now, it might seem shocking to try to work with the Euler product, in either form, without infinite sets (the reals themselves - Dedekind cuts - are infinite sets), but an infinite sequence, speaking informally, need not imply an infinite set, as long as the rule is finite. What's generated, then, provided the series is convergent, is a Dedekind cut, which would be a proper class under the axiom of Tarski finiteness. However, that's not really important, but rather, what's important is the rule, and manipulations of this rule under the arithmetic operations by a sort of composition; a rigorous definition not found in this post would most likely come from the theory of computation.

I remember how much easier I found vector calc than I did linear algebra. This was because in vector calc, I had already guessed most of the basic operations from simply generalizing the rules from one-dimensional high school calculus. At that age, I couldn't wrap my head around the fact that this clearly wasn't enough. To some extent, modern set theory's treatment of analysis is a formalization of this misapprehension, so that it ceases to be a misapprehension. Let's take the alternative perspective, Turing standing on Peano's shoulders rather than Zermelo's for the hypercomputability hierarchy to replace Gödel's. Let's enhance our confusion to create a grand certainty.

Back to the point. To recap, the "evidence" is that if you take the logarithmic derivative of the zeta function you get the logarithm of each prime in turn divided by one minus that prime to the negation of the parameter, which can alternatively be expressed, per Euclid, as the logarithm of that prime multiplied by the sum from zero to infinity of one over the powers of the prime in question to the parameter (remember, though, that this is a finite rule as opposed to an infinite set). Considering that, all terms being positive, this converges absolutely, the terms can be rearranged; note that the terms that are generated by the rule are exactly fractions with prime powers raised to the parameter in the denominator with the log of the base in the numerator, so let's put them in the order of the prime powers.

Before we go on l let's go back to that sentence, "considering that this converges absolutely, the terms can be rearranged." Remember when we all learned that in high school? But the concept of a permutation on an infinite set is taken as read there, so it's imperative to prove the equivalent principle again in this new framework. By the definition of limit, we can always run the partial long enough that it'll be within some given distance of the sum. In that, provided we have some idea where the terms are going, there must be a maximum destination, which must be equal or greater. Therefore, it must include all the terms, and more. If they're all positive, then this can only be larger. Therefore, it's true of the absolute value. Term-by-term summation following from that of the partials and the arbitrarily low upper bound on the tail, you get from there and the convergence of a subseries to the case of absolute convergence.

Anyway, back to the "evidence." From here it's pretty clear. You can make this sum by layering up integrals that start at each prime power, which will each be the parameter times the log (when there is one) divided by the index to the power of the parameter plus one. So moving the sum inside the integral (since it's absolutely convergent - same argument as before, only this time with Riemann sums slid in) you'll get the second Chebyshev function divided by the parameter of the integral to the power of one plus the parameter of the zeta function, all multiplied by the parameter of the zeta function. If you plug in one, the derivative of the log of the zeta function blows up, and so this integral blows up as well - which is what you'd expect if the second Chebyshev function asymptotically approached identity, because then you'd be dividing identity by the square to get the reciprocal so that would blow up. You'd also expect, then, that subtracting the reciprocal would cause it not to blow up, and in fact this would imply the asymptotic approach. The suggestion of this comes from the fact that if you multiply the zeta function in the log by one less than the parameter, the derivative manifestly converges.

So that's the "evidence," but it's not the proof because we can't obviously move the limit inside, not even with the normal machinery of set theory/analysis. So why can we move the limit inside? The short and incoherent answer is because there are no zeroes on the edge of the critical strip. The proof of this I won't restate - it's just algebra and trig - but the connection still isn't inherently obvious. From there, what's left is the Wiener-Ikehara theorem, which it's even more imperative to view now in terms of Fourier analysis, that is to say, constructing a function from an uncountable accumulation of sinusoids. Might be some kinks to work out there in the absence of infinite sets.

So basically, as I said before, the actual proof is based on the notion of an "approximation of unity," a family of functions of integral one that approach zero bar an infinite isolated point. (Again, none of these concepts exist, but speaking in shorthand, that's what it is.) Multiplying this by offsets of another function allow you to show that that function approaches a constant, and in this case to show that the second Chebyshev function approaches the identity. The fact that this approaches a limit comes from the equation discussed before.

Now, so far, I'm just drunkenly ruminating on cud already so thoroughly chewed. What's interesting, though, is that not only are all these limits determined by finite rules, but that they themselves are their solutions, and these rules for arithmetic operations axiom schemata, rather than inferences. The question, then, isn't whether they follow, but whether they're eliminable in whatever context is important to our purposes. And that's what brings us back to Euclid.

Remember, the challenge there is to build from the sparse postulates of his geometry and "number theory," to which not even he can hold himself entirely, to get to such a radical conclusion regarding the prime numbers. In his terms, as before, it can be expressed only in terms of the harmonic series - at least without great difficulty. New chapters might be introduced bringing it to modern terms (i.e., at the very least, terms in which RH in prime-number form would be meaningful) without a modern notion of infinity, but these would be lengthy chapters. For now, let's go with the prime counting function and the harmonic series. I said then that we would need definite error bounds, but let's replace that with "as close as we like."

Let's work backwards. We first need to start with the fact that an increasing (non-strictly) function that's convergent when you add it up (at integers because no infinity - note that this is an integral of a function of countable range in the ordinary paradigm) subtracting identity and dividing by the square must approach in ratio identity. This follows from the divergence of the harmonic series in effect - if it didn't approach identity in ratio, then the ratio would either approach a number greater or less than one, or it would diverge, being unable to oscillate due to the function's monotonicity. If it did those things, the function wouldn't converge.

From there, two things remain to get us to "as close as we like." If only I could remember what they were. (I suppose I should mention on this point that my hippocampus is a pickled seahorse on a toothpick.) I suspect they're to show that the increasiness is finite for positive parameters and that it reflects the limit. The former is easy enough to show from series (to wit, by saying that it holds for any power, however small, the concept being expressed easily enough geometrically), the latter coming from Fourier analysis. In any case, really, there's no real logic to this other than what I mentioned before, only the relevance of it to this notion of mathematics without infinite sets.

So let's not fuck around. Fourier analysis, the analysis of periodic functions on the real line expressed as the integral of an uncountable set of sine waves, without infinite sets. How? Well, not at all, really, but basically by geometry, obviously. Sine waves, after all, come from sines, the string of a bow the chord, the musical sense of "chord" after all coming from the chord a string made against a lyre (as far as you know), and Euclid knew the law of cosines in a geometric, very Alexandrian form (propositions twelve and thirteen of book II).

So let's go back to the very beginning of this blog. The normal distribution. Now, the slovenly proof I gave then doesn't really show anything and isn't the normal (so to speak) approach anyway. A better notion of probability comes through set theory and measure theory, the latter of which certainly doesn't work as normally understood in a finitist paradigm.

So you'll remember I explained (ish) why a series of fair coinflips should approach the standard normal, but I didn't explain probability beyond that at all. So what is probability? Well, you take a set, a subset of that set's power set closed under complementation (relative to the base set) and countable union, and a function from the latter to [0, 1] such that the empty set maps to zero, the union of countable disjoint sets maps to the sum of those sets' image (implying the union of countable non-disjoint sets is less than or equal to the sum of those sets' image), and the whole shebang maps to one. That's a probability space. Now, if the set has to be finite, you're golden to model, e.g., cards or dice, but you're SOL if you want to model a dartboard, unless you feel like working it out at the quantum level, but even there there may or may not be infinities where probability is concerned. Therefore, under this paradigm, the base set and space would have to be understood in terms of the rules that generate them - but aren't they anyway?

So with that in mind, the notion of approaching a probability space via a series of fair coinflips is analogous to the notion of approaching a real, computable or uncomputable, by a Turing-equivalent machine. Now, I didn't bring up Turing machines (exactly) above because even in this paradigm uncomputable numbers can be defined, such as the sum of two to the power of the negations of successive busy beaver numbers.

I've been writing this for months now, and I'm sure it's wicked self-contradictory and flows like tar, but fuck it, it's going up.

#infinity #set theory #probability #gambling #computability #prime number theorem

3 notes · View notes

valeriequinonez1 · 6 years

Text

Sensibilities and Intuitions of the Master Designer; an Interview with Cecil Balmond, part 2

[Images courtesy of Balmond Studio]

Last summer, BUILD met with engineer-architect-artist, Cecil Balmond at his London Studio to discuss his most recent projects and the thinking behind his experimental design process. Prior to opening Balmond Studio, his career spanned 40-plus years at Ove Arup & Partners where he worked on pioneering projects with renowned architects all over the globe. Balmond discussed the notion of architecture in a dynamic environment, the designer’s intuition, and his most recent projects. For part 1 of the conversation, hop over to ARCADE Magazine, Issue 36.1, available in print and on their website.

Tell us about your previous role as Deputy Chairman at Arup, where you led thousands of engineers and architects. There were seven of us on the board of directors at Arup and I was head of building business globally with around 6,000 people under my supervision. When I joined, Arup was a company of about 5,000 people and when I left it was 11,000 people. The job was a huge bureaucratic task in one way, but on the other hand, I was the only director who had an active design group. My design group ranged from 25 to 60 people and we handled about 30 jobs per year. I would choose two or three of these projects and I’d personally lead the design.

It was at this time that I began setting up the Arup architectural practices in Beijing, Shanghai, and Turkey, as well as the sector architectures such as ARUP Sport and Arup Health. Arup Sport was a great success and we hired expert architects to lead the projects, like Richard Rogers and Norman Foster.

How would you characterize the spirit at Arup? Arup was a special organization because really it was led by Ove Arup at the beginning, who was a philosopher and a mathematician more so than an engineer. He was a man of the world with open ideas. That way of being really filtered down to certain people, like myself and others, who, if I’m honest, believed in design, and not necessarily engineering or architecture. Design was a much wider thing to us. Architecture had its own expert skill zone, and when it comes to the real grit of architecture, the specifications, window schedules, and the engineering, there is a horrendous, humungous amount of calculating to be done. But those are the mechanical parts of it. A great engineer is simply wonderful to watch at work because they’re intuitive, and I don’t just mean structural engineers, but environmental engineers, lighting engineers, etc. They’re dealing with intangibles almost, and yet they have an intuition that influences the building in a very holistic way. This method of working significantly contributed to my thinking that there are no limits in design.

Was there a particular moment or project that encouraged you to formalize your practice as an engineer-architect-artist? No, it’s like a lot of things in life, you drift. It’s a question of being an opportunist. Occasions occur where your instinct is primed to take advantage of key opportunities. If you are a creative person, you are pushing, not knowing what you are pushing at and then something comes up and you just jump, you take it, and I think my career has been a series of those jumps.

There was a cathartic moment at about age 35 when I was smoking outside my office and decided when I went back in I could never do the same thing again. It was that decisive, I just knew. But I didn’t know what was next. So, I went back in and threw out all my learning and started learning again. I went through a personal mentorship for the next five years. I studied at night, going back to the original treaties of mathematics, going back to the three forbidding books and six postulates of the Greek mathematician Euclid. I went back to the very first precepts set by the Greeks, like the philosophy of the point above a line, above a plane, the line being drawn through thepoint, above the plane, being parallel, and so on. The books written about those postulates engaged my mind totally. It provided me with a mobile sense of geometry. Those postulates soon led to the idea of proportion.

The next step took five to ten years and it involved believing in a mobile sense of geometry, where forms are constantly in motion and architecture is only a snapshot in time. This led to a proportional sense of space and ultimately an episodic treatment of design. This sequence was dependent on releasing my hand and thinking more freely. It required that I start thinking differently about design, that buildings don’t stop at the four corners, and that they don’t necessarily have to have a floor, a roof, and sides. It was a personal odyssey of unlearning and it is key to the work I’m doing today.

Is there a common way that you approach each design project? The way I work is generally scale-less as an idea. I tend to start with a metaphor or a feeling, something really vague. Then comes a sketch of something in space, some notion of space, or more accurately the notion of the intersection of space between it, it’s interiority, and the relation of the context of where it is. Just purely conceptually and it’s nothing to see yet. It might just be a few lines or a blotch. Then comes the idea of what is it. Is it art, engineering, or an architecture piece? Then comes the functionality, then comes the choice of scale. Once you choose scale, the material locks in. If it’s very small its thread or wire. If its humongous, its steel or trusses. Then comes configuration of scale. Last of all would be structure — actual structure as it means to an architect today. The actual skeleton, the actual thing is the last thing. If you start with that at all, you’ve lost the building. You’ve lost the spirit, you’ve lost what the building can do. At the end of my book Informal, there is a very interesting table of the hierarchy of decision making that goes through my mind.

You note that challenging assumptions is critical to your work. What is a recent example where challenging an assumption made a significant difference to the outcome of the project? Toyo Ito and I designed the 2002 Serpentine Gallery Pavilion together and we decided to start with a box. Upon looking at a map of London’s Hyde Park, where the Pavilion is located each year, we realized that the park is a collection of crisscrossing lines. Then came the idea that this pavilion is the gathering of lines. We started playing around with algorithms and the type of geometries similar to the movement of a ball around a billiard table until we hit upon a geometry that came back on itself and completed the box. This exercise allowed us to break the boundaries of the envelope and challenge the notion of the box. Even though it was a 50-foot by 50-foot structure, the viewers inside had no idea that they were in a box. Spatially, it was much bigger than the bounding box of its geometry.

Tell us about your discovery of aperiodic tile invented by the mathematician Robert Ammann. 20 years ago, I felt that architects and the graphic arts had no idea what mathematics does, so I started researching numbers. I quickly realized that the prime numbers have powerful sequences that are unpredictable. They look like a kind of music when I interpret them, and they’ve held my interest for years. The geometry of these tiles is based on the prime numbers and this is what makes them aperiodic in that their assembly results in a new pattern each time — they never repeat. Daniel Libeskind and I applied the tile to the V&A Spiral which is the proposal for an extension to the Victoria and Albert Museum in London.

Your QXQ project addresses the need for prefabricated, modular housing. In your experience, what are the hurdles of implementing prefabricated, modular housing on a mass scale? It’s the biggest challenge in the industry and no one’s cracked it — not even Arup. Years ago, they went in with a huge contractor here in the U.K. who does housing and they spent a lot of money researching prefab design. The result ended up looking like every other prefab. And that’s the problem, because in the end, for mass production, you need corners and right angles, and once you have corners and right angles, to save money you close the surface and then you’ve got a box. You can go and cut corners and triangles out and make it look interesting, but it’s still a box. You haven’t cracked the sense of living.

In order to be successful, prefabrication shouldn’t start with conventional ideas. It would be great to think that prefab housing could inject a new idea of living in such prescribed spaces. No one has been successful at this yet and I tried a bit with the QXQ project. So many boxes have already been done and I don’t want to do another box. What can I add to it apart from cuteness and your sensibility of design? I was interested in refuge housing and wanted to investigate low technology, using my ideas to make things less expensive. I wanted to try to use architecture in adaptable ways using cheap materials but highly sophisticated design techniques to make an interesting statement while being functional. My design started with a dodecahedron and sliced off parts. This allows stacking in any direction and, interestingly, it created the idea of a colony of tightly fit modules rather than a collection of prefabricated homes. All the sudden, you’re into biomorphic design and while the architecture and structure are straight-forward, the services become challenging. Where do the ventilation, water, and sewer systems fit? We haven’t quite cracked it yet. We’re building two units as a test, but we really need to build 12 of them to check our assumptions, and we need to be building hundreds of units to be commercially viable. There are a number of interested clients from all over the world and a particular army was interested in 40,000 units. That’s the kind of scale we need to make the concept great, but we need to get the first one right.

Rem Koolhaas cites that, “through your work, engineering can now enter a more experimental and emotional territory.” Are academic engineering programs following your lead? I know certain architecture and engineering programs have taken my books as curriculum. The Scandinavians were the first to take up Informal, then some universities in the States and in England started using the book. I think it’s impacted young architects more than the engineering community as I suspect that the engineers may be enticed by the work but are afraid to pick up the book because the thinking is so radically different.

How has a non-linear approach to design affected the other areas of your life? I started organizing parts of my practice at Arup in a non-linear basis and it was very successful. Rather than applying top-down thinking, I began using an informal, emergent thinking. As an example, I deliberately don’t file my books, so I go searching my library and randomly pick a book, and then open up to the middle of the book and I read. That immediately kicks me into something I never even thought of. In the early ‘90s I became convinced that the world was non-linear. We simply fight it to be linear in order to understand it. But actually, it was not understandable in the first place.

You’ve had a synergistic relationship with artist Anish Kapoor, including your collaborations on the 2003 Marsyas exhibit at the Tate Modern, the Temenos sculpture in north England, and the Arcelormittal Orbit built for the 2012 summer Olympics. Tell me a bit about the balance you two have found working together. Anish and I came together originally for the Marsyas exhibit at the Tate Modern. It’s not so much the mechanics of the form making with Anish, it’s more about the discussions we have of what does it mean. I think that’s the driving spur between us. The mechanics of how you make the form is part of whoever’s skill set it falls under. So, if the items involve big spans, I’m doing it. If it’s an issue of color and surface, he’s doing it. Creative tensions about what is good or not arise, but it’s precisely these discussions that lead to the power of the form. It’s about a visceral reading of the form and how it moves you physically.

In any of these designs, you’ve got non-linear architectures and engineering forms, but it seems like you’re typically able to use a standard kit-of-parts like steel channels and I-beams. Do you feel that the materials and parts ever limit the form factor? No, because I always take the materials as a given out of pragmatism rather than thinking that I’m going to invent a new material or form. This isn’t to say that you compromise what you’re doing, but you need to rationalize how you’ll build a design and in that comes certain decisions to make about the material.

Do you have any structural inventions that you’re particularly proud of? The roof of the Arnhem Centraal project in the Netherlands includes a giant column that’s approximately 100-feet wide. It twists in space to support the roof and ground floor planes and it’s one of my best inventions. I thought the design would be prohibitively expensive, but it wasn’t.

from Civil Engineering http://blog.buildllc.com/2018/06/interview-cecil-balmond-part-2/ via http://www.rssmix.com/

#interior design

1 note · View note

serenova · 7 years

Text

Humans are Weird

So I’ve been reading a lot of those “Humans are Weird” posts having to do with us and Aliens, and I really love them.

But there’s one thought I’ve had that I haven’t seen other’s talk about before. So here goes.

===============

Humans have developed a comprehensive understanding of the workings of the universe before being able to send themselves to other planets. Now that doesn’t seem strange to humans, because, well, they did it.

But what if an alien race didn’t have the same kind of geniuses humans have? What if there was no equivalent to Einstein, to Reeman, to Gauss, to Newton, even Euclid.

Humans have explored the universe with just their minds. Imagining first 4 dimensions and then 7, 8, even 10. They’ve developed string theory. quantum mechanics, have scientific theories that are decades, even centuries away from being testable. But they keep exploring, keep thinking, keep imagining, keep yearning to know more. They even developed nuclear fission AND fusion before space flight!

And what if this TOTALLY weirds an alien species out? They’re basically the opposite of humans. Yeah, sure they invented interstellar travel, but for completely different reasons. They were simply hungry for resources. They mined their systems asteroid belt, they built great cities but they didn’t ponder the beginning of the universe, they didn’t figure out it’s age until centuries after they first reached for the stars.

Imagine a human trying to explain their species fascination with the unknown to another species that doesn’t have that same yearning.

For all we know it might be something we might never be able to understand about each other.

We may be the Doc Brown’s of the universe, but it may be only for the sheer fact that we ask, “What’s beyond? How far can we go? Where does the universe start and end? How did we get to here?” And other species aren’t nearly so curious about the universe.

Half of the discoveries made my mathematicians about how geometry works is them going “I wonder if....” or physiciscts going “Hey, what about...?”

And I think that’s really amazing.

#humans are weird #humans and aliens #what are even humans #humans are doc brown

1K notes · View notes

udemy-gift-coupon-blog · 5 years

Link

Learn Trigonometry and It's Application+Hundreds of Examples ##FreeCourse #ApplicationHundreds #Examples #Learn #Trigonometry Learn Trigonometry and It's Application+Hundreds of Examples In a sense, trigonometry sits at the center of high school mathematics. It originates in the study of geometry when we investigate the ratios of sides in similar right triangles, or when we look at the relationship between a chord of a circle and its arc. It leads to a much deeper study of periodic functions, and of the so-called transcendental functions, which cannot be described using finite algebraic processes. It also has many applications to physics, astronomy, and other branches of science. It is a very old subject. Many of the geometric results that we now state in trigonometric terms were given a purely geometric exposition by Euclid. Ptolemy, an early astronomer, began to go beyond Euclid, using th~ geometry of the time to construct what we now call tables of values of trigonometric functions. Trigonometry is an important introduction to calculus, where one studies what mathematicians call analytic properties of functions. One of the goals of this book is to prepare you for a course in calculus by directing your attention away from particular values of a function to a study of the function as an object in itself. This way of thinking is useful not just in calculus, but in many mathematical situations. So trigonometry is a part of pre-calculus, and is related to other pre-calculus topics, such as exponential and logarithmic functions, and complex numbers. The interaction of these topics with trigonometry opens a whole new landscape of mathematical results. But each of these results is also important in its own right, without being "pre-" anything. We have tried to explain the beautiful results of trigonometry as simply and systematically as possible. In many cases we have found that simple problems have connections with profound and advanced ideas. Sometimes we have indicated these connections. In other cases we have left them for you to discover as you learn more about mathematics. Triangle is basically a geometric structure of any existing phenomenon having three sides and three angles. Application of triangle has vast area of interest in everyday life. The description of this course is very practical and emerging. All the concepts are described logically to make the well understanding. The sequence of all videos are categorized in step by step learning. After the basics concept of triangle I have described the complete description of trigonometry. I have tried to put each contents related to trigonometry in this course. I hope that the students will enjoy this course Who this course is for: Students who have problems in understanding of triangle, their types and how to solve triangle? 👉 Activate Udemy Coupon 👈 Free Tutorials Udemy Review Real Discount Udemy Free Courses Udemy Coupon Udemy Francais Coupon Udemy gratuit Coursera and Edx ELearningFree Course Free Online Training Udemy Udemy Free Coupons Udemy Free Discount Coupons Udemy Online Course Udemy Online Training 100% FREE Udemy Discount Coupons https://www.couponudemy.com/blog/learn-trigonometry-and-its-applicationhundreds-of-examples/

0 notes

sarajhines · 5 years

Text

What Physics Extended Essay Topics to Prefer?

On the off chance that you are one of those understudies who chose to commit their lives to the specific sciences, you'll without a doubt see the genuine estimation of one of these subjects:

• Can octocopters become the model of future flying vehicles?

• What is the Oberth Effect in astronautics?

• When and how does rainbow show up?

• Can you clarify the wonder of the ACE quality?

• How does roar show up?

• What specific angles can impact gravity?

Interesting Chemistry Extended Essay Topics

Almost certainly, fewer understudies would lean toward this subject on the grounds that not every one of them is genuine scientific experts skilled right now. Is it accurate to say that you are one of them? Peruse on!

• Analyze all advantages of the natural nourishment for a patient's wellbeing.

• Is it conceivable to create nourishment without pesticides?

• Can rankle drag out the timeframe of the realistic usability of certain items?

• What are the key side effects of nutrient insufficiency?

• What sort of nutrients, common or engineered, impact wellbeing better?

• Is it conceivable to change yeast into a helpful biofuel?

Math Extended Essay Topics for Goal-Oriented Students

• Why is it so significant for each individual to contemplate math?

• Who was the genuine creator of polynomial math?

• What apparatuses and strategies would it be a good idea for us to use to improve the information on geometry in secondary schools?

• Analyze the commitment of Archimedes and Euclid in the advancement of math?

• Discuss all the positive and negative parts of Venn Diagrams.

• What online programming items can assist understudies with upgrading their aptitudes in math?

• Review crafted by notable mathematicians.

What is the Easiest English Literature Extended Essay Topics?

On the off chance that you are an imaginative individual and venerate perusing, you'll unquestionably like one of the subjects beneath:

• How does J. Baldwin's grandstand prejudice develop in his books?

• What is your disposition toward present-day authors? Is it accurate to say that they are worth consideration?

• Read one of the books composed by E. Bronte and clarify how she grandstands religion.

• Read completely any novel, get one character, and expound on their changes.

• What procedures did American authors use with the plan to portray nature in their books?

Valuable and Easy Economic Extended Essay Topics

• Can the significant expense of oil influence the economy of the nation?

• The issue of detainees in your nation.

• Can hostile to tobacco law diminish the number of guests in cafés?

• Can migrants improve the advancement of the cultivating business?

• Cash advance credits and their impact on the nation's economy.

Who Can Help you?

As should be obvious, the conceivable theme for your all-inclusive exposition is sitting tight for you to discover and investigate it! Try not to worry over this undertaking when you have endless more to finish, for your IBDP as well as your school. We've just done a large portion of the activity for you. What's more, you know, you should confide in our custom composing site! Simply hit us up on our live visit or through email, and our prepared specialists will expeditiously go to your guide. Good karma!

0 notes

douchebagbrainwaves · 4 years

Text

HERE'S WHAT I JUST REALIZED ABOUT TIME

What made oil paint so exciting, when it first became popular in the fifteenth century, was that you can start a startup on less money than most people think. Hard as this was to believe in the mid twentieth century servants practically disappeared in rich countries, and the reactions that spread from person to person in an audience are always affected by the reactions of those around them, and the reason why, unlike other languages, Lisp has dialects. And yet the trend in nearly everything written about the subject is to do the landscapes in the backgrounds of Italian paintings. Even Einstein probably had moments when he wanted to have a cup of coffee, but told himself he ought to finish what he was working on first. You can do this if you want to work in and it's something people are likely to soon. If it didn't suck, they wouldn't have had to make it prestigious. The SFP was just an experiment to get things started. You can even use it tactically.

It's not enough to make it open. More generally, you can build a whole programming language. That's the recipe for success in big companies, or in most schools. In an opera it's common for one person to write the music. In the mid to late 1980s, all the pressure is in the sciences, true collaboration seems to be vanishingly rare in the arts, and particularly in oil painting. In 1960, John McCarthy published a remarkable paper in which he did for programming something like what Euclid did for geometry. This is particularly true of young people who have till now always been under the thumb of some kind of authority. I'm still not entirely sure they're correct.

But finally I've figured out how to express this quality directly. Conversely, the extreme version of the two we aim at. Just as a speaker ad libbing can only spend as long thinking about each sentence as it takes to hear it. Even people sophisticated enough to know what they want. Why is that so? And while it would probably be a good speaker. This doesn't mean you have to be doing something you not only enjoy, but admire. I'm designing a new dialect of Lisp. He showed how, given a handful of simple operators and a notation for functions, you can have a fruitful discussion about the relative merits of Ford and Chevy pickup trucks, that you couldn't safely talk about with others. We should fix those things. Then you want to do, you have to figure out what's actually wrong with him, and treat that.

Y Combinator described as an incubator. Design usually has to be in a position to pick and choose among projects. Now that we know what we're looking for in metaphors. Design means making things for humans. Much as we disliked school, the prospect of an actual job was on the horizon. Always produce is also a heuristic for finding the work you love. This test is especially helpful in deciding between different kinds of academic work, because fields vary greatly in this respect. Gone were the mumbling recitations of lists of features. It would not work well for a language where you have to spend all your time working. This is one way I know the rich aren't all getting richer simply from some new system for transferring wealth to them from everyone else. It's painful to keep them apart, because it's painful to observe the gap between them.

As far as I know, without precedent: Apple is popular at the low end and the high end, but not unfair. Occasionally the stimulation of talking to a live audience makes you think of new things, but in general this is not going to try to explain in the simplest possible terms what McCarthy discovered. I could say that force was more often used for good than ill, but I'm not sure. That was probably part of the reason I laughed so much at the talk by the good speaker at that conference was that everyone else did. Maybe, I suggested, he should buy some stock in this company. The 20th best player, causing him not to make the most money are those who aren't in it just for the money. Just as a speaker ad libbing can only spend as long thinking about each sentence as it takes to hear it. All the best hackers I know are gradually switching to Macs. What I really want is to have good ideas, but in general this is not going to generate ideas as well as writing does, where you can spend as long thinking about each sentence as it takes to say it. Sometimes judging you correctly is the end goal. If the rich people in a society got that way by taking wealth from the poor, then you have the destination in sight you'll be more likely to work than attacking wealth in the hope that you will thereby fix poverty.

And there are other topics that might seem harmless, like the relative merits of Ford and Chevy pickup trucks, that you couldn't safely talk about with others. In high school she already wanted to be a doctor. And even within the startup world, there has simultaneously been a huge increase in individuals' ability to create wealth. But it was also something we'd never considered a computer could be: fabulously well designed. It might be a good thing for the world if people who wanted to get rich. No one should be. In practice, to get good design if the intended users include the designer himself. As hard as people will work for money at a time. That idea is not exactly novel. Other times it's more unconscious. Three months' funding is enough to get into second gear.

#automatically generated text #Markov chains #Paul Graham #Python #Patrick Mooney #test #world #audience #times #painting #speaker #job #libbing #practice

0 notes