#logarithm base

[brandboosterx]

In this video, we delve into the art of solving logarithmic equations with different bases, demystifying the process for you step by step. Whether you're a student brushing up on logarithms or someone facing more complex problems, we've got you covered. Understanding how to work with different bases is crucial when faced with logarithmic equations. We break down the techniques, providing clear explanations using frequently used words to ensure that you grasp the concepts effortlessly. No more getting stuck on those seemingly perplexing logarithmic problems!

[fleshwizard]

Map of the Known Universe

[felikatze]

For every liter squared of water there is about 10 to the power of -14 mol squared water ions (H+ and OH-) which is referred to as the constant Kw. And because the concentration of H+ and OH- is about equal, both have a concentration of 10 to the power of -7.

(This already supposes the simplification of concentration = activity)

The negative logarithm to the base of 10 in this case would be 7. In the case of the concentration (c) of H+ ions, this is shortened to pH. The counterpart to describe the c of OH- ions is, then, the pOH value.

This is why when the pH and the pOH value are added together, they always make 14. In the case of water, where the concentration of H+ and OH- is equal, both values are 7. Values differ depending on the concentration of acids and bases.

In strong acids and bases, which are said to dissociate entirely into their ions (though that is not entirely correct), it can be said that the pH-Value is, as before, entirely equal to the concentration/activity of H+ ions (negative logarithm to base 10, ofc).

In weaker acids and bases, I am still figuring it out, but it sure does involve my friend the Massenwirkungsgesetz

[saintkevorkian]

[sejdkvinna]

shells for witchcraft

Abalone Shells

Abundance, peace, and compassion

Abalone shells are very colorful, and the insides are coated with the same material that make up pearls, nacre.  These colors are said to be incorporating all the colors of the sea and sky.  In addition, the iridescent colors promote a feeling of abundance, peace, and compassion.  In addition, the colors can represent the upper chakras, the Abalone shell is useful for developing psychic powers.  Since Abalone shells have also been a food source for thousands of years, Abalone shells represent the giving power of the Ocean.

Abalone shells can also be made into jewelry and amulets, and not only will they promote abundance, peace, and compassion, but they will also impart a calming, spiritual vibration to help the wearer through emotional difficulties.  In addition, Abalone shells have also been historically sacred in Polynesian and Native American cultures, especially as a vessel for sage while smudging.

Scallop Shells

Travel and spiritual quests

Symbol of Venus and Water

Scallop shells are distinguished from other shells by the multiple rays and ridges, which all converge at a single point near the base.  To early humans, this likely symbolized the sun's rays converging on the horizon at sunrise or sunset.  Because of this, Scallop shells are a good symbol of Venus, also known as the Morning Star or Evening Star.  Later, to medieval Christians, the Scallop represented the path of pilgrims to holy sites.

Continuing, Scallop shells had a sacred meaning in Christian Europe from as early as the first pilgrims.  The pilgrims, although mostly illiterate, were able to be guided to the pilgrimage sites with the Scallop symbol.  In addition, the Scallop is a symbol of the Saint James the Apostle, and the symbol of the Scallop can be found through countless Christian altars and cathedrals.  Because of the association with pilgrimage, the Scallop is a symbol for those who have undertaken a spiritual quest.

In addition, the Scallop shell is often depicted as the seat or carriage of ocean deities in art.  For example, in art Poseidon often rides upon one, and Aphrodite is seen rising up from the sea in a Scallop shell.  In addition, Scallops are often on Pagan altars to symbolize ocean deities and the element water.  Because of this, Scallops are often used as a chalice, offering bowl, or vessel for love magick.

Nautilus Shells

Renewal, growth, and rebirth

The Nautilus travels hundreds of miles in the currents of the sea, and can be considered the nomad of the sea.  In fact, the fitting name means "sailor" in Greek.  More importantly though, the Nautilus shell is an important symbol of mathematics and sacred geometry because of the logarithmic pattern in its spiral.

This shape is because unlike other shell creatures, rather than shedding its shell and finding a larger one, the Nautilus increases the size of the shell.  The Nautilus shell grows to create a larger chamber for the body as it grows.  Because of this, the Nautilus shell is a good symbol of growth, renewal, and rebirth.  Additionally, the outwards spiral trend of the Nautilus suggests that it can continue growing indefinitely.  Meditate with a Nautilus shell or use it in your magick for spiritual growth and expansion.

Auger Shells

Protection and aggression

Auger shells are distinguished by its shape, which is an elongated spiral that has an opening at the base and a point at the tip.  In the language of shells, the Auger is considered to be both masculine and feminine energetically, therefore representing completeness.  There are hundreds of varieties of Auger shells all around the world as well.

In addition, Auger shells come from a type of predatory sea snail, and the aggression of the animal is implied by its narrowly focused shape.  Auger shells have a venomous tooth as well, making it a good shell to use in hexes and curses.  In magick, Auger shells are one of the only shells that are associated with the planet Mars.  Other uses for Auger shells are for headdresses, magick wands, and protective charms.

Cowrie Shells

Fertility, luxury, and wealth

The Cowrie shell is characterized by its yellow or white color and egg-shape, with two rows of teeth along a gap in the center of the shell.  Cowrie shells are admired for their natural polish, which symbolizes refinement and luxury.  In fact, the Italian word for Cowrie is porcellana ("little pig"), which the word porcelain is derived from.  Because of its shape and polish, Cowrie shells have been used as currency, decoration, and religious items in nearly every part of the world.

In addition, because of its shape, the Cowrie shell symbolizes the vulva.  Therefore, the Cowrie shell is often used in charms regarding wealth and fertility.  Cowrie shells can also be strung onto necklaces or sewn into garments.  Additional uses for Cowrie shells include divination.  There are many different systems of divination with Cowrie shells, mostly stemming from African and Afro-Caribbean cultures and traditions.

Conch Shells

Awakening and mermaid connections

Conch shells have been symbolically used by cultures around the world since ancient civilizations.  The conch (or Shankha) is used by Tiben Buddhists as a way to call people to worship, and the conch shell is also seen in the hand of the Hindu god, Vishnu.  In addition, the sound of blowing through a conch is similar to the sound of AUM (the sound of the universe), which has an awakening effect (similar to chanting OM).

In addition, the conch shell is often shown in the hands of mermaids, which makes the conch shell a good tool for connecting to mermaid energy and mermaid spirits.  Similarly, blowing through the conch shell can be used as a tool for summoning mermaid (and other ocean creatures) spirits.  Meditating with the conch shell can be a way to get in touch with the ocean and ocean spirits, since you can hear the ocean in the conch shell.

Sources:

https://www.groveandgrotto.com/blogs/articles/the-magick-of-seashells

https://zennedout.com/magical-uses-meanings-of-shells/

[bobacupcake]

Can you do one about the Sea of Thieves water?

OK

so . there was a biiig long talk about this at siggraph one year!! you can watch that here if you'd like . in the time between me getting this ask and me fully recreating the water, acerola also released a great video about it . the biiig underlying thing they do and the reason why it looks so good is they are making a Really Detailed Ocean Mesh in realtime using something called an FFT (fast fourier transform) to simulate hundreds of thousands of waves, based on a paper by TESSENDORF

WHAT IS AN FFT - we'll get to that. first we have to talk about the DFT - the discreet fourier transform. let's say you have a SOUND. it is a c chord - a C, an E, and a G, being played at the same time. all sounds are waves!!! so when you play multiple sounds at the same time, those waves combine!!! like here: the top is all 3 notes playing together, so they form the waveform at the bottom!!

now if someone handed you the bottom wave, could you figure out each individual note that was being played? how about if someone handed you a wave of One Hundred Notes. you would think it would be very hard. and well, it would be, if not for the Discreet Fourier Transform.

essentially, there is a way to take a bunch of points on a waveform comprised of a bunch of different waves, add them all together, do some messed up stuff with imaginary numbers, that will spit back out at you what individual waves are present. i made a little test program at the start of all this: the left are the waves i am putting into my Big Waveform, the top right is what that ends up looking like, and all the little rainbow points on it are being sampled to spit out the graph at the bottom right: it shows which frequency bands the DFT is finding (here it is animated)

this has enormous use cases in anything that deals with audio and image processing, and also,

THE OCEAN

tessendorf is basically like, hey, People Who Are Good At The Ocean say that a buuuunch of sine waves do a pretty good job of approximating what it looks like. and by a bunch they mean like, hundreds of thousands to millions. oh no.... if only there was a way we could easily deal with millions of sine waves..........

well GREAT news. not only can you do the DFT in one direction, but you can also do it in REVERSE. if you were to be given the frequency graph of a noise for example, you could use an INVERSE DFT to calculate what the combined wave graph looks like at any given time. so if you were to have say, the frequency graph of an oceaaaan, for example, you could calculate what the Ocean wave looks like at any given time. and lucky for us, it works in two dimensions. and thats the foundation of the simulation !!!!!

BUT WAIT

as incredible as the DFT is, it doesn't scale very well. the more times you have to do it, the slower it gets, exponentially, and we are working with potentially millions of sine waves here

THE FAST FOURIER TRANSFORM here we are . the fast fourier transform is a way of doing the discreet fourier transform, except, well, fast. i am Not going to explain the intricacies of it because its very complex, but if you want to learn more there are a ton of good 30 minute long videos on youtube about it . but essentially, due to the nature of sine waves repeating, you can borrow values as you go, and make the calculation Much faster (from exponential growth to logarithmic growth which is much much slower, and scales very well at higher numbers). it's, complicated, but the important part is it's so much faster and the diagram kind of looks like the shadow the hedgehog story plot

so if we use the inverse FFT on a graph of a rough estimate of what frequency of waves in the ocean (called a spectrum, basically tells us things like how many small waves, how many big waves, how different waves follow the wind direction. sea of thieves uses one called the phillips spectrum but there are better ones out there!!) now we have our waves !!!!!!! we can also use another inverse FFT to get the normals of the waves, and horizontal displacement of the waves (sharpening peaks and broadening valleys) through some derivatives . yayy calculus

OK MATH IS OVER. WE HAVE OUR WAVES!!! they are solid pink and look like pepto bismol. WHAT NOW

i cheated a bit here they look better than not being shaded because i am using the normals to reflect a CUBEMAP to make it look shiny. i think sea of thieves does this too but they didnt mention it in their talk. they did mention a FEW THINGS THEY DID THOUGH

FIRST OFF - SUBSURFACE SCATTERING. this is where the sun pokes through since water is translucent. SSS IS REALLY EXPENSIVE !!!!!! so they just faked it. do you remember the wave sharpening displacement i mentioned earlier? they just take the value where the waves are being sharpened and this will pretty naturally show off the areas that should have subsurface scattering (the sides of waves). they make it shine through any time you are looking towards the sun. they also add a bit of specular ! sss here is that nice blue color, and specular is the shiny bits coming off the sun. the rest of the lighting is the cubemap i mentioned earlier, i dont know if thats what they use but it looks nice !!!!!

then the other big thing that they do is the FOAM !! sorry i lied. there's more math. last one. you remember the wave sharpening displacement i Just mentioned. well they used that to find something called the JACOBIAN and well im not even going to begin to try and explain what it means but functionally what it does, is when the jacobian is NEGATIVE it means waves are clipping into eachother. and that means we should draw some foam!!! we can also blur and fade out the foam texture over time and continuously write to it to give it some movement, and bias this value a bit to make more or less foam. they do both of these!!!

YAYYYYY !! OK !! THAT'S SEA OF THIEVES WATER!!!!! THANKS FOR WAITING ALL THIS TIME. you can see my journey here if you would like to i have tagged it all oceanquest2023

thank you everyone for joining me :) i had fun

[transhuman-priestess]

I've been wanting to try this film for a while. It's Lomography Fantôme Kino, also available as Wolfen DP31.

It's an ultra-low-speed Black And White film. It has an ISO of 8. Not 800, not 80, 8. So it's got incredibly high contrast, incredibly long exposure times, and it looks bitchin.

Film speed is logarithmic, every time you double the number, it doubles the sensitivity. Your standard walkabout film back in the day was 400, decently good for outdoor photography and indoor with flash. Your phone camera now is usually around ISO 800 at a minimum if you're indoors. (As a side note, ISO is not a unit, it's a rating. It's not correct to say a film has "8 ISOs")

Based on the above, this film requires about 7x as much light as my phone to make a good picture. This lets you slow down your shutter speed even in bright daylight to get long-exposure shots like the one in Row 3.

I'm surprised it shot so well with flash indoors.

I'm definitely buying this film again. It fucking rules.

P.S. I traded in my old canon A2 for a slightly newer Canon EOS 33 and it slaps. These were all taken on that one.

[aspaceinthecosmos]

Happy International Women's Day!

Women are underrepresented in the fields of astronomy and physics. According to the International Astronomical Union (IAU), between 20-30% of astronomers are women. While many well-known astronomers are men, there have been numerous female astronomers in history who have made incredible discoveries, but who history has forgotten. Today we'll go over some of those women and their accomplishments.

Annie Jump Cannon (1863-1941)

Annie Jump Cannon is the woman responsible for our current stellar classification system, which organizes stars based on spectral types and temperature.

She worked at Harvard Observatory as a computer, working on the Henry Draper Catalogue, which attempted to map and classify all the stars in the sky. She was regarded as the best out of the computers, being able to accurately classify the stars incredibly quickly, up to three stars per minute.

Cannon's classification system (O, B, A, F, G, K, M) is still in use today, and separates stars into one of these spectral groups based on different characteristics of their absorption lines.

Henrietta Swan Leavitt (1868-1921)

Henrietta Leavitt is most well known for her discovery of the period-luminosity relationship of Cepheid variable stars.

Henrietta Leavitt was also a computer at Harvard Observatory in the late 1800s and early 1900s, working on cataloguing positions and luminosities of stars. In 1912, Edward Pickering published a paper with Leavitt's observations, which contained a relationship between the brightness of the Cepheid and the logarithm of the period of it.

This discovery, and the ensuing P-L relationship (sometimes known as Leavitt's Law), allowed astronomers to determine the distance to further objects. Because Cepheids are visible in nearby galaxies, astronomers were able to determine that these galaxies (or nebulae, as they were called then), were actually much further away than previously thought, leading to our current understanding of the universe and galaxies outside our own.

Cecilia Payne-Gaposchkin (1900-1979)

Cecilia Payne-Gaposchkin was the first astronomer to conclude that stars are primarily made of hydrogen and helium.

At the time her thesis was proposed in 1925, it was thought that the sun had a similar elemental composition as the Earth. Payne-Gaposchkin, however, had studied quantum physics, and recognized that the differences in absorption lines between different stars was due to ionization and temperature differences, not elemental differences, and that stars were primarily made of hydrogen and helium, with heavier elements making up less than two percent of stars' masses.

Her theory was met with resistance, and she even put a disclaimer in her thesis, saying the results were "almost certainly not real" in order to protect her career. She was, however, proven right within a few years, and her discovery shaped our knowledge of the composition of the universe.

Vera C. Rubin (1928-2016)

Vera Rubin is most well known for studying the rotation curves of galaxies, and finding a discrepancy that didn't align with the current understanding of physics. This discovery was used as evidence of dark matter, as proposed by Zwicky in the 1930s.

Rubin discovered that spiral galaxies didn't rotate as expected. When looking at our solar system, the outer planets orbit slower due to the inverse square nature of gravity. However, this decaying rotation curve wasn't what was found in spiral galaxies - rather, the outer edges of the galaxies were rotating at about the same speed as the inner areas.

According to Rubin's calculations, galaxies contained 5-10 times more mass than what was accounted for with visible matter. This supported the dark matter theory, and resulted in the current "anatomy" of galaxies, with the visible matter surrounded by a dark matter halo.

Jocelyn Bell (1943-present)

Jocelyn Bell discovered pulsars among a sea of data as a graduate student at Cambridge.

Pulsars (shortened from pulsating radio stars) are rapidly rotating neutron stars, which emit bursts of radiation at extremely short and consistent time intervals.

Bell discovered these, and published the findings in a paper with her thesis supervisor, Antony Hewish. in 1974, Hewish received the Nobel Prize in physics for this discovery, while Bell was omitted, due to her status as both a woman and a graduate student. In 2018, she was awarded the Breakthrough prize in Fundamental Physics for her discovery, and used the three million dollar reward to help minorities in physics.

[sokkas-therapist]

Zukka incorrect quote based on a conversation my friends had in our trig class today

Zuko: I can’t figure out how to make this logarithmic function hit these points on the graph

Sokka: oh…I just made linear lines to hit the points

Zuko: but that’s cheating??

Sokka: no it isn’t

Zuko: it literally is- the point of the assignment is to create logarithmic functions, you can’t just use linear lines.

Sokka: well if I hit you over the head with a pan-

Zuko: WHAT?? why are you hitting me in the head, I thought you loved me???

Sokka, continuing as if Zuko hadn’t interrupted him: -and you pass out, then it wouldn’t matter what type of pan it was, would it? You would still be unconscious.

Zuko: what the fuck is wrong with you

Sokka: a graph is a graph.

Sokka: …

Sokka: also I do love you

[starset-mnqn]

Broke: N's birthday is March 14th, AKA 3/14, because that's 3.14. Pi! The math thing! Teehee. All math nerds get pi birthday :)

Woke: N's birthday is February 7th, AKA 2/7, which correlates to 2.7, the first two digits of Euler's number. The base for NATURAL logarithmic functions! Which has applications in probability theory, which N has canonically expressed some interest during the Summer Nights & Wishing Stars event in Pokemon Masters. Astrologically speaking, this would make him an Aquarius. This fits his character due to his unusual mannerisms and compassionate nature towards Pokemon. Furthermore, if we set his birth year to 2000, (designated as the International Year for the Culture of Peace and the World Mathematical Year) then he would also have his moon sign be Pisces, which helps to tie into his goal of fulfilling his dreams and furthers his compassionate nature...... *I am forcibly dragged off the stage*

[tarrarre]

having natural logarithmic sex with my exponentially based wife

[ladysaytenn]

For YEARS I've been trying to find a solution to low-maintenance toon shading in Blender that also keeps the overall look, but I've been hitting a wall every attempt.

I recently watched this video from Visual Tech Art making a post-processing shader in Unreal 5 that put the lighting information through a logarithm operation, rounding the results, and squaring it back up. This inspired me to create a similar implementation in Blender's compositor, and....

...it was that easy all along????

This is just Viewport Compositing by the way, which only supports the Combined Pass right now. It does require tweaking based on the look you're going for, and you can tweak it even more with render passes and custom masks, but by default it looks pretty nice! Subsurface scattering and bounce lighting are still accounted for as opposed to just using the Shader to RGB node, and it can work with Cycles!

[vedalkensamurai]

For a given complex number c, what is the behavior of the iterative sequence z_(n+1) = e^(z_n) + c, where z_0 = 0 and e is the base of the natural logarithm? Experimenting with a few random values, I noticed that the sequence often converges, either to a single value or a repeating loop of values. I wondered if there was any pattern to how quickly it converges, so I wrote a program to systematically check.

In the image above, each pixel represents a complex number ranging from -5.12 - 5.12i to 5.12 + 5.12i, with a distance of 0.04 between neighboring pixels. At each point the program iterates until it reaches a value it's seen before at that point (within a tolerance of 10^-10), then colors that point accordingly: red if it reaches it very quickly through magenta if it gets there slowly, and black if it never repeats in 256 iterations. (This is much slower to calculate than the Mandelbrot set, since it needs to check each z against all previous zs, whereas the Mandelbrot set only checks against a constant escape threshold.)

The other question I had about these sequences was, when they do converge to loops, how long are those loops? My early experiments showed a variety of periods. So of course I wrote a program to check the period lengths too. In the image below, the brightest purple corresponds to single-point convergence, and as the loops get longer the purple gets darker. Once again, black is where no repetition was found. Here I only did 128 iterations per point to save time, so the black area is a bit bigger than above.

I'm interested in the borders between the different purples, as it looks like there is some complexity in the shapes there. I'll be doing some zoomed-in images of those areas next.

[xyymath]

The Magic of Euler’s Identity

Mathematics has its moments of sheer beauty—when abstract concepts collide to form something elegantly simple, yet profoundly deep. Euler's Identity is the pinnacle of this beauty. Expressed as: eiπ+1=0e^{i\pi} + 1 = 0

It may seem deceptively simple, but it links five fundamental constants in mathematics:

e - Euler’s number, the base of natural logarithms (~2.718).

i - The imaginary unit, defined as −1\sqrt{-1}.

π - The famous constant, representing the ratio of a circle's circumference to its diameter (~3.1416).

1 - The multiplicative identity.

0 - The additive identity.

Why is Euler's Identity so Special?

At first glance, Euler's identity seems almost trivial—a relationship between numbers that don’t seem to belong together. Yet, it’s a striking result of complex analysis and trigonometry. The formula brings together exponentiation, complex numbers, geometry, and even calculus in a single equation. How is this possible?

Euler’s identity emerges from Euler’s formula, a fundamental equation in complex analysis: eix=cos⁡(x)+isin⁡(x)e^{ix} = \cos(x) + i \sin(x)

For any real number xx, this formula expresses the complex exponential eixe^{ix} as a combination of cosine and sine functions. When x=πx = \pi, we get: eiπ=cos⁡(π)+isin⁡(π)=−1+0i=−1e^{i\pi} = \cos(\pi) + i \sin(\pi) = -1 + 0i = -1

Thus, Euler's identity follows immediately: eiπ+1=0e^{i\pi} + 1 = 0

This result connects imaginary numbers, trigonometric functions, and exponentiation in an astonishingly compact form.

Why Do Mathematicians Love It?

Euler’s Identity is often called "the most beautiful theorem in mathematics." Why? Because it encapsulates the interplay between several areas of mathematics in one concise equation. It’s elegant, unexpected, and remarkably simple—an almost magical blend of different mathematical elements.

Mathematicians love it because it challenges our intuition. We tend to think of numbers like ee, π\pi, and ii as separate entities, each with its own mysterious properties. But Euler’s identity reveals that they are inextricably linked, suggesting that seemingly unrelated areas of math have hidden connections.

Some Mind-Blowing Examples and Connections

Trigonometry and Geometry: Euler’s formula gives rise to the unit circle in the complex plane. As eixe^{ix} traces a path around the unit circle, it maps directly onto the sine and cosine functions, giving a visual and geometric interpretation of trigonometric identities.

The Complex Plane: In the complex plane, eixe^{ix} represents a point on the unit circle with an angle xx from the positive real axis. This brings an entirely new way to think about rotations, using exponential functions.

Quantum Mechanics: Euler’s identity has practical applications in quantum mechanics. The wavefunctions in quantum physics often involve complex exponentials of the form eiθe^{i\theta}, linking Euler’s identity to the fundamental laws governing particles.

Differential Equations: The beauty of Euler’s formula shines in solving differential equations. In systems that oscillate (like springs or circuits), Euler’s formula simplifies the solution process and makes the math elegant and tractable.

[ell-arts]

PMATGA Headcanons: Surnames

This thing has been a headscratcher for as long as I have been a fan of the show. Or it might be just me-

But I've always struggled to imagine what some of the characters' surnames/last names could be. Especially the main trio. Mostly because it's tricky to come up with surnames that would fit them, and something in line with the show's running theme of naming things after geometry or character-specific quirks.

We know that Stratos' last name is Spheros (derived from stratosphere) and thus it's also Betrayus' surname. We know of Mr Strictler, who only revealed his surname and never his first name, so we can deduce that Sherry's last name is Strictler (she's his daughter). Then there's Mr Dome, Ms Globular, Sir Cumference, etc. But other than these characters, we never really heard of any other surnames mentioned in the show.

There is Spheria Suprema, and hey it could be probable that that is her surname, but I'm leaning a bit more towards the idea that it's her celebrity name/Pac-Pong champion title/nickname rather than her actual given surname. The chances of Spheria being given a name that rhymes with a descriptive title such as 'Suprema' at birth is a little slim if you ask me (this is just my rationalisation, don't come at me xD)

So that leaves us with the rest of the cast. I've seen some really creative attempts from the fandom to give the characters their surnames. So much so that I can't really come up with any good ones in comparison xD

But hey, these are the best I could come up with, along with some headcanons or reasons for why I chose them. Subject to change!

-

Pac Orbon (Yellow Orbs, duh...)

Cylindria Eden (based on the Garden of Eden, and sounds hippy-ish)

Spiral Logarith (based on the logarithmic spiral in math)

Elliptica Spheros (her mother kept her last name)

Sir Perimet Cumference (his first name is based on perimeter (of a circle), which can be shortened to 'Peri' but he still prefers 'Sir C')

Trayus Rotundin Spheros (he changed it to Betrayus Sneakerous Spheros as part of his rebellion-turned-revolt)

Zac Orbon (again, yellow orbs...)

Sunny Orbon (maiden name was Sunny Solari - a play on sol and solar, from the sun - which is also round and yellow)

Spheria Solari (tends to go by Spheria Suprema, her celebrity name, since that's what many people know her as)

Specter Eidolon (greek for 'idol' or 'phantom')

Blinky "Shadow" Scarlos (based on 'scarlet')

Inky "Bashful" Whimson (based on 'whimsical')

Pinky "Speedy" Chacier (Old French for "chase/hunt")

Clyde "Pokey" Oren (just based on orange, but can also have different meanings depending on the language)

[The Ghost Gang's names are a combination of lore that I've found relating to the original Pac-Man games. For example, Inky's original name in the retro games was Bashful, and his Japanese name is kimagure, which means 'whimsical', hence Whimson.]

[otesunki]

where can i find information on {0|0}. on *. i'm obsessed

colossal infodump incoming

alright there's this Very Very Good Book called Winning Ways for Your Mathematical Plays which explains combinatorial game theory and how they link into surreal numbers in what can only be described as an Unreasonable level of detail, including how it ties into Surreal Numbers oh God surreal numbers is just as loaded of a term Okay let me take a brief detour into That

so surreal numbers are like. imagine defining every number as a pair of sets of other numbers. a set of numbers Less than it and a set of numbers Greater than it. and we write this as foo = {less than foo|greater than foo}. so like. 0 = {|}, 1 = {0|}, 2 = {1|}, 1/2 = {0|1}, -1 = {|0}, that type of thing

the reason why surreal numbers are mindmelting is the fact that they happen to be the Largest Totally Ordered Proper Class and they contain Literally Every Other Totally Ordered Number System inside of them. totally ordered here means that any two numbers are related by < > or =. so like. complex numbers and quaternions aren't included. surreal numbers also behave just like real numbers in that you can do arithmetic on them exactly how you'd do arithmetic with any other real number. and for the surreal numbers that are also real numbers the classic laws of commutativity and associativity hold.

however the surreal numbers are. a.

a bit bigger than just all the real numbers. because there are also numbers that are infinitely big or small and you can make infinitely-bigger-than-infinte numbers and the arithmetic operations still work and also have you ever wanted to take the logarithm base infinity of a number too bad that's defined now

games are what happens when you look at the surreal numbers and go that's for rookies and decide that actually yes {1|-2} makes sense what are you Talking about. why Can't on = {on|} that's a perfectly sane definition also over = {0|over} also Also actually you can define a number that's Even Closer To Zero than over is and if you churn through the calculations you can literally Prove that 0 < tiny < over Yes the number is called tiny

right okay so what is star? in fact if x = {A|B} then -x = {-B|-A} and because A and B are both just the set { 0 } and 0 is {|} you can trivially prove that negating star gives you back star, it's not greater than zero because it has zero in the right hand side, it's not less than zero because it has zero in the left hand side, and it's not Equal to zero because playing a game with a value of zero means that the first player Loses but a game with a value of Star means that the first player Wins which means neither player has an advantage but it's a balanced game in a different way to how 0 is balanced and this is reflected in the Thermograph of star which is a way to draw what star looks like when you Heat it up and Heating a number basically means moving its left and right hand sides closer together in value until eventually they meet up

so Zero because both of its sets are empty is just a vertical line at x=0 because it can't heat up nor cool down but Star on the other hand stays at 0 when it is cooled down but if you Heat it up (make t negative) then actually the two zeroes in it begin Diverging and going in opposite directions

but because the whole thing is Horizontally symmetric it means that the number is its own negative because negating a number is equivalent to a horizontal flip. also if you thought thermographs couldn't get more complicated and involved you're wrong

so. That's. what. star is.

so have you ever wondered what would happen if you had a bunch of numbers that are all mutually recursive and not defined in terms of anything el-