#approximation

[approximationlocater]

APPROXIMATION LOCATER

[whats-in-a-sentence]

Whether or not you actually follow its moment-to-moment development, according to classical physics you can talk about the past and future, in principle, with a confidence that is controlled by the detail and the accuracy of your observations of the present.¹

1. It is well known that the equations of classical physics cannot be solved exactly if you are studying the motion of three or more mutually interacting bodies. So, even in classical physics, any actual prediction about the motion of a large set of particles will necessarily be approximate. The point, though, is that there is no fundamental limit to how good this approximation can be. If the world were governed by classical physics, then with ever more powerful computers, and ever more precise initial data about positions and velocities, we would get ever closer to the exact answer.

"The Fabric of the Cosmos" - Brian Greene

[literarycatchall]

“I still can’t say what life is for, but it can’t be to pretend

that every part of it is knowable, or that what appears to be

to the naked eye or in the middle ground or documented on paper

approximates a person any better than a daisy does our sun.”

— “Habitual Nebula,” Timothy Donnelly (2023)

[uncertain-noob]

The Nearly Regular Heptagon From Classical Times

So, there's this construction that goes back at least hundreds of years, and maybe thousands, of a figure that is very nearly a regular heptagon. I've done half of the construction in the geogebra screen shot below.

So, you start with an isosceles 4-9-9 triangle, and draw its circum circle. Here, that triangle is on the vertices p1, p2 and p5. Those are going to be vertices on 'regular' heptagon. So, if you really believed that, that we were making a regular heptagon, you would just walk your length 4 side around circle, ie create E as the second point on the intersection between the big circum circle and the circle centred at p2 with radius 4, then make another radius 4 cricle at E to get D and then you're done with that side.

But the actual construction is way more elaborate. It takes the line from p2 to p5, and puts a point on it, here labeled B, that is distance 4 from p2, then draws a line from p1, through B, and where it intersects the circumcircle, that's another vertex, here labeled D. And then E is on the bisector of p2 and D. So, there are things going on here that make a kind of sense. If the vertices were evenly spaces around the circle, then p1-p2-B would be congruent to p5-D-B and they would be isosceles. this construction, they are still both isosceles and the distance from D to p5 is so close. Its 4.0089. we want it to be 4, so the error is less than 1 part in 400. The distances from p2 to E and from E to D are even closer: the error is less than 1 part in 800. So its a great attempt, and its something we know to be impossible without folding, so, bravo. But there is no way that anyone who knew what they were doing is going to claim this is a construction of a regular heptagon, right? like, the fudge is huge. And its obvious! Why aren't you constructing the radius of the circumcircle for some chosen side, or vice versa, and then just walking the side around the circle? You know you're fudging the construction as soon as you draw the point on the diagonal, or at the very least, you're guessing you might be. In fact, if you construct vertices that are exactly 4 from p5 on the circle, and draw the final two vertices by bisection, you can calculate all the side lengths using Ptolemy's theorem and you can see that there's more length on the sides you construct with bisection, and that extra length is still going to be there however you construct the other vertices. On the other hand we have the very sophisticated construction of Archimedes which uses a currently shunned technique called neusis. He very clearly understands the complexity, and overcomes it, just not with a straight edge and compass.

[tenth-sentence]

Moreover, the theory intimates that the familiar notions of space and time do not extend into the sub-Planckian realm, which suggests that space and time as we currently understand them may be mere approximations to more fundamental concepts that still await our discovery.

"The Fabric of the Cosmos" - Brian Greene

[katsinspats]

I think my copy of the game is broken they've been doing this for 30 minutes

Crop of the Biolizard edit I did bc it makes me laugh:

[reasonandfaithinharmony]

PSA: You can never have too many photos or videos of your pets. I mean this sincerely.

[chiropteracupola]

everyone will look at my fish hat in an expeditious style

(pattern from here, slightly modified to be longer so that there's lots of room in the tail for hiding packets of fruit snacks)

[whats-in-a-sentence]

In fact, when physicists began to measure force and acceleration with precision, they found that if exactly twice the force was applied to a particular body, then exactly twice the acceleration was achieved*; if exactly n times the force, then exactly n times the acceleration.

* Well, not quite exactly. Einstein's relatively introduces a correction that is vanishingly small under ordinary circumstances, but this essay is devoted to the Newtonian approximation and I am ignoring, for now, the Einsteinian approximation.

"The Stars in their Courses" - Isaac Asimov

[chloesimaginationthings]

SOTM Tiger rock reminds me of Golden Freddy,,

[blobbei-art]

Amalgamation doodles compilations! Despite the many heads, he is surprisingly fun to draw

[goodpix2021]

Coming

Soon enough. We are a people in a hurry. I don’t know why, but we are. We are still living in winter for a few more weeks, but people act like we have moved on. Nature has a funny way of dealing with that, the northeast had about a billion inches of snow dumped on it this week. More is on its way. Down on the farm we had about a foot of snow and we live in Central Virginia. Even though we…

[minyard-05]

hands down the funniest scene in the king's men is when neil is like "what do i have to do to get you to play exy without actively attempting to injure people" and andrew, lazer fucking focused responds with "take off your shirt" like MY MAN. YOU ARE NOT SUBTLE

[oh-no-its-bird]

Has anyone done this yet? Someone has to have done this already, right?

[mischieviem]

Hello Dean