Ideas at an Exhibition

mathematical explorations

[fckyeahnetart]

Sabey Data Center. NYC. https://www.wired.com/2017/01/enormous-places-humanity-stashes-data

[curiosamathematica]

Let’s play the following “chaos game”. Consider a regular triangle and take some initial point in it. Now pick one of the three vertices at random, connect it with the current point, and mark this line’s midpoint as the new point. If you repeat this process many times, what will the resulting shape look like? You might expect to find the point simply shattered everywhere, without much structure, but surprisingly, the points will in fact create a Sierpinski fractal.

There are lots of variations on this theme, some of which can be found on this Wikipedia page.

Animation made in Mathematica.

[chaosophia218]

Unveiling the Mandelbrot Set. Back in the 1970s and 1980s, mathematicians working in an area called dynamical systems made use of the ever-advancing computing power to draw computer images of the objects they were working on. What they saw blew their minds: fractal-like structures whose beauty and complexity is only rivalled by Nature itself. At the heart of them lay the Mandelbrot set, which today has achieved fame even outside the field of dynamics.  The Mandelbrot set is a fractal. Fractals are objects that display self-similarity at various scales. Magnifying a fractal reveals small-scale details similar to the large-scale characteristics. Although the Mandelbrot set is self-similar at magnified scales, the small scale details are not identical to the whole. In fact, the Mandelbrot set is infinitely complex. Yet the process of generating it is based on an extremely simple equation involving complex numbers.

[thewriterkb]

Not explaining science seems to me perverse. When you’re in love, you want to tell the world. My lifelong love affair with science.

Carl Sagan, The Demon-Haunted World: Science as a Candle in the Dark

(via blunt-science)

[arsmathematica]

In mathematics, as in most human pursuits, we draw pictures not only as a means to represent reality but often as a means to explain reality. The distinction I make is that one may draw a picture of a building to be beautifully exact or one may draw a schematic that encodes relative size, angles, materials etc. without paying attention to ornamental details. We call the latter ones doodles, cartoons, diagrams, or sketches. 

Above are doodles from Michael Spivak’s Calculus on Manifolds. It’s a great little introduction to differential geometry for anyone with solid calculus and point-set topology. These aren’t exactly life changing pictures, but the book was published in 65 and I just dig these groovy blue drawings.

These images should be treated as copyrighted by Spivak.

The first picture is a representation of a smooth function sending the blue square on the left to the interior of the black squiggle on the right. We like to think of smooth functions as simply pushing  the points in a plane around a bit. This figure was used in the proof of the inverse function theorem, which basically tells us that a small enough area around a point (point “a” in this case) smooth functions don’t really do anything.

The next picture gives an example of changing of variables . In reality we might like to think about cardinal directions (the blue square on the left), but then be able to switch to interstate directions (the squiggly one in the middle), but then be able to switch to state route directions (the squigglier one on the right). 

The third picture is a drawing of the definition of manifold. Basically if you look closely enough at a manifold it looks flat. 

The next picture is used in the discussion of surface area. Where for curves (bits of string) we know that the length (if you pull it taught) is just the upper bound on the length of line segments that approximate it (using a short measuring stick). This is called rectification. This picture is meant to show that no such similar thing can be done with surfaces (sheets of paper). That is, all polygonal (flat) approximations can be increasing in surface area. 

The last figure is an example of a star convex set, meaning every two points are connected by two straight lines (it looks like a star). The statement of Poincare’s lemma is included. But it’s a special statement about a certain homology theory on contractible spaces. You can think of it as indicating that that which goes on in the interior blue region is determined by that which goes on on the star’s edge.

[lbroland-blog]

See pg 133 for answers

[graftheory]

Platonic Solids by Moni

[mathhombre]

Mystery Curve

John D. Cook shared this great curve: e^{it} – e^{6it}/2 + i e^{-14it}/3 from the book Creating Symmetry: The Artful Mathematics of Wallpaper Patterns by Frank Farris.

I converted it to parametric, and was immediately wowed by the possible variations. Give it a try!

[curiosamathematica]

Structures are the weapons of the mathematician.

Nicolas Bourbaki, the collective pseudonym under which a group of (mainly French) 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality. Their work led to the discovery of several concepts and terminologies still discussed. (via curiosamathematica)

[eka-mark]

just realized that if i used homotopy type theory to prove a result about memory evolutive systems, it would be a HoTT MES

[argumate]

HoTT damn!

[everything-narrative]

homotopy type theory is HoTT shit.

[abramovicinstitute]

This animated GIF is 20 frames long and takes 2 seconds to repeat itself, but if you pick any point and follow its motion continuously it will take 2000 seconds (about 30 minutes) for the point to return to the same location it was in when you started watching it! How long can you follow a single point?

Mathematica code:

G[p_, q_, a_, b_, c_, t_] := Graphics[  Table[   {Opacity[.7],    Disk[     {.71*Cos[a (.05*t + n)*Pi/500 + p],      Cos[b (.05*t + n)*Pi/500 + q]},      .002*Cos[c (.05*t + n)*Pi/500] + .005]},   {n, 1, 1000}],  PlotRange -> {{-.715, .715}, {-1.005, 1.005}}, ImageSize -> 500] Manipulate[   Show[    G[0, .2, 29, 13, 11, t],    G[0, .2, 19, 31, 9, t],    G[.3, .29, 3, 23, 17, t],    G[0, .3, 43, 7, 3, t],    G[.1, .13, 31, 19, 2, t]], {t,1,20,1}]

[idease]

Interesting

[idease]

Of course, not everyone prefers the austere joys of algebra to the earthy pleasures of geometry. Algebraic topologists thrill to categories, functors and natural transformations, while geometric topologists like drawing pictures of hideously deformed multi-holed doughnuts in 4 dimensional space. It's all a matter of taste.

John Baez

[mathprofessorquotes]

Isn’t it refreshing not to use a calculator? Doing math by hand cleanses your spirit. It’s wholesome.

Calculus professor (via mathprofessorquotes)

[twocubes]

dun dun DUN

[twocubes]

the details of this thing are incredible though; like, on the (Groupoids) side, the morphisms from G to H are actually spaces equipped with left- and right-actions by (completions of) G and H, and the 2-morphisms are morphisms between them that respect these actions.

at the same time, on the topos side, every hom-category can be identified with the category of models, in the source topos, of the theory that the target topos classifies.

every geometric theory has a classifying topos. this means the categories of groups, rings, fields, modules over a given ring, small categories, objects in a topos, real numbers, monoids, semigroups, finite projective planes (etc) can all be seen as hom-categories of (Toposes), and thus by the equivalence, can be seen as categories of spaces equipped with actions.

:V

[idease]

Animated Lissajous figure generator

The page shows two main inputs: M and N. Imagine a single dot, moving periodically back and forth. M is the speed of this horizontal motion. The dot is also oscillating up and down; N controls this vertical speed. Combine these two perpendicular movements for cool shifting sieves.

[sub-division-blog]

[curiosamathematica]

The theory of groups is a branch of mathematics in which one does something to something and then compares the results with the result obtained from doing the same thing to something else, or something else to the same thing. This is a broad definition but it is not trivial. The theory is a supreme example of the art of mathematical abstraction. It is concerned only with the fine filigree of underlying relationships; it is the most powerful instrument yet invented for illuminating structure.

James Roy Newman, The World of Mathematics (via curiosamathematica)