#math visualization

I started to craft a visualization model for a Pascal"s 4-simplex.

Now I need to insert the 12 triangle disks. (I need to puzzle them into each other like a kind of interwebbing. )

Pictured above, we have the ordinal spiral, depicted using “matchstick notation.” The ordinal spiral is a beautiful and intuitive way to represent the structure of the ordinal numbers.

Quoting Wikipedia, “In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct natural numbers. Ordinal numbers are thus the ‘labels’ needed to arrange collections of objects in order.”

Note that “each turn of the spiral represents one power of omega.” Omega is, in turn, the first infinite ordinal. I’d rather not get into the technical details of how omega is defined. Instead, I’d like to encourage you to take a moment to appreciate it’s arithmetic and algebraic structure.

As a kid, I recall being endlessly obsessed by the idea of “being able to do math with infinity.” And there it is... stunning, isn’t it? Aren’t you tempted to factor out that omega in the middle of the matchstick, two turns in (from the middle-right)? Do it! Go ahead, you’re free now! Just be sure to thank my homie, Georg Cantor. 

Pascal"s triangle

I used transparent/translucent paper to color the numbers of pascal"s triangle.

Each color represents a prime number:

Red: 2

Orange: 3

Yellow: 5

Green: 7

Blue: 11

Violet: 13

For each layer (and the according prime number) I colored the numbers of pascal"s triangle if the pascals triangle number is divisable by that prime number.

The advantage of this transparent/translucent paper I used:

You can hold multiple layers against the light (and see the overlapping colors):

2 and 3:

^For instance: if you look at the 6 in the 5th row it is a combination of red and orange - 2 and 3 - as 6 is built up by the primes 2 and 3 (2*3)

2 and 3 and 5:

3 and 5 and 7:

Recently I posted a slightly different version of this stellated octahedron, but I changed some things like the vertices" names to letters resembling the accompanied color: ( - for instance, the outer vertex of the red tetrahedron is "R".)

My stellated octahedron wire model:

The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers.

It is the simplest of five regular polyhedral compounds, and the only regular compound of two tetrahedra. It is also the least dense of the regular polyhedral compounds, having a density of 2.

It can be seen as a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two overlapping equilateral triangles, centrally symmetric to each other, and in the same way the stellated octahedron can be formed from two centrally symmetric overlapping tetrahedra. This can be generalized to any desired amount of higher dimensions; the four-dimensional equivalent construction is the compound of two 5-cells. It can also be seen as one of the stages in the construction of a 3D Koch snowflake, a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure. The first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron.

[Copied from Wikipedia:]

Rotation of the stellated octahedron - gif animation: