… with intersections like those of a Kepler-Poinsot polyhedron.

From

#####A NEW RHOMBIC HEXECONTAHEDRON #####¡¡ PDF file – 118·11㎅ !!

by

#####Branko Grünbaum .

I don't want the meaning just how is called and if it's possible to learn it on a book or a video

Tell me if there is something i could improve

Hello, fellow mathematics entousiath.

I made some "fractal" drawing using python which led me to some questions regarding convergence toward similar picture with a different set of rules. In particular, is it to be expected ?

When I was in class, I always drawn the most boring recurring serie :
Start with an isoscele right triangle, then from its hypothenuse draw a new right triangle where the lenght of the side is half the previous hypothenuse. Repeating this process results in the following pattern.

Basic pattern

Now that I am a lazy adult, I used python to extend the formula to draw additional spirals (with same orientation) which starts from each exterior of the original spiral. (I used this process recursively which includes the new drawn spirals. A small detail is that the basis of each spiral is a replication of the previous triangle rather than an other homothetie).
As a result, we get the followings for the firsts steps (I don't know exactly how to define a step since its a mix of recursion and loop, respectively for branchs creations and deepness of a spiral).

https://preview.redd.it/mba51lhn3nsc1.png?width=640&format=png&auto=webp&s=e841ac5248710a0854d8bafae6605c0fce383337

Finally, we can extand the process to infinity. In practice, I stop when a length of a triangle is smaller than 2 pixels. The result kind of look like a Pytagorean tree (or a Lévy C curve, which I know nothing about).

https://preview.redd.it/g6cob4bf4nsc1.png?width=640&format=png&auto=webp&s=e36d8866650e73e2714cba8fe0b341fd72a921be

The original purpose was to cover the whole plane, which is a replication of the figure rotated by n*pi/2, a total of 4 time:

https://preview.redd.it/uka017vx4nsc1.png?width=636&format=png&auto=webp&s=6c37c7ea46f8e2a680bf8dbd594066f56dde41be

In hindsight, it's surprising to realize that the resulting pattern resembles a Pythagorean tree. When you take a step back, you know than the main constituant are isosceles rights triangles and a downscaling by a factor sqrt(2)/2. Additionaly, the individual spirals have broadly the same shape that branches in the pytagorean tree.
However, can we anticipate a convergence toward a similar drawing, considering the differences in rules and basic constituents ? I'm not well-versed in fractals, so perhaps this is a trivial matter.

Additionally, are there methods available to verify if the outcomes are truly identical? Or is it too complex to find suitable metrics for comparing the Pythagorean tree with this particular construction?
Futhermore, is the drawing impacted by the resolution, maybe adding steps would result in a different drawing ?

Thank you for your time, if you have some to help me satisfy my curiosity !

PS : I used python and matplotlib, the coloring is a fortunate artefact of the function imshow, which do some kind of interpolation, and use viridis as default color map, hence the green.

Hi all, it looks like this hasn't been asked in at least a year -- at least, searching "software" doesn't show a result in the last year.

So I'm wondering which software has which advantages for making math animations. Although my current interest is specifically about animation, it might be useful to others to have a more sprawling conversation about making vizualizations more generally.

I currently know about the following.

Manim

Plus: makes beautiful videos, has active community and support.

Minus: A bit slow and takes up computer resources like memory and time.

Beamer which converts to GIF

Plus: Use familiar LaTeX and TeX commands, makes a PDF which can be convenient. Can compile the pages into a GIF and clear up space.

Minus: Less pretty, and the end-to-end process still takes a while.

Geogebra

Plus: WYSIWYG, fast, does not eat up a lot of space, quite pretty.

Minus: Missing some things you'd typically want, not as flexible. For instance, can't mark a length in geometry with a curly brace in a very easy way. I think you'd have to insert an image and then place it. Can't intersect 3D solids, only surfaces. So on.

Matplotlib

Plus: Powerful, fast.

Minus: Takes a lot of learning and getting used to. Not easy to insert text and formulas.

If anyone knows more tools that compete with these, I'd love to hear it!

my partener (and i) got the answer as 1, but notes from an acquaintance state it's 2. what is it, finally?

I wanted to do this since the start of high school. I've recently graduated & got good enough to use trig/polynomials/algebric equations to design Pakistan's flag

Images from

#####North Dakota State University — Erdős–Rényi random graphs #####¡¡ PDF file – 1·34㎆ !!

See also the closely-related

#####North Dakota State University — The giant component of the Erdős–Rényi random graph #####¡¡ PDF file – 1·26㎆ !!

& the seminal paper on the matter - ie

#####P ERDŐS & A RÉNYI — ON THE EVOLUTION OF RANDOM GRAPHS . #####¡¡ PDF file – 1·14㎆ !!

The department of random graphs has actually been one in which a major conjecture was recently established as a theorem - ie the Kahn–Kalai conjecture. Here's a link to the paper in which the proof, that generally astonished folk with its simplicity, was published.

#####A PROOF OF THE KAHN–KALAI CONJECTURE

by

#####JINYOUNG PARK AND HUY TUAN PHAM .

TbPH, though, I find the sheer matter of the proof - ie what it's even a proof of - a tad of a long-haul even getting my faculties around @all ! It starts to 'crystallise', eventually, though … with a good bit of meditating-upon, with a generous admixture of patience … which, I would venture, is well-requited by the wondrosity of the theorem.

It's also rather fitting that its promotion to theoremhood was within a fairly small time-window around the finally-yielding to computational endeavour of the

#####ninth Dedekind № .

This is actually pretty good for spelling-out what 'tis about:

#####Threshold phenomena for random discrete structures ,

by

#####Jinyoung Park .

This business of random graphs is closely-related to the matter of percolation thresholds , which is yet-another über-intractible problemmo: see

#####Dr. Kim Christensen — Percolation Theory #####¡¡ PDF file – 2·39㎆ !!

, which

#####this table of percolation thresholds for a few particular named lattices

is from. It's astounding really, just how intractible the computation of percolation thresholds evidently is: just mind-boggling , really!

From

####Two packing problems #####¡¡ 136·25㎅ !!

by

####Vojtech Bálint .