#4th dedekind number

[art-of-mathematics]

The task I wanted to do:

- task 1: Visualize the fourth dedekind number

What I needed to do in order to do task 1:

- task 1.1: Draw a hypercube and

- task 1.2: then give each of the 16 vertices a name - which is the 16 different "cells" (I dunno which word to use, sorry) of a venn diagram with 4 overlapping sets plus a circle around that 4-venn diagram.

- - -- ---

//additional info: For the 3rd dedekind number visualization I already posted I needed to use 20 cubes. Each of the 8 vertices of a cube was related to 3 sets - A, B and C - and the intersections of these sets. (AB, AC, BC, ABC)

You remember this post:

... and these details:

...and now back to the 4th dedekind number, which follows the same principle, - but in 4 dimensions.

- So we need a hypercube - and a 4th set/letter:

Now we have 4 letters - alias 4 sets

- and 11 combinations of these sets:

AB, AC, AD, BC, BD, CD, (6)

ABC, ABD, ACD, BCD, (4)

ABCD (1)

=> 11 different intersections + 4 "pure" sets. (sorry for ignorant wording)

So, we have 15 of these now. What is with the 16th (as the hypercube has 16 vertices)? Yeah, that one is the circle you imagine around the 4-set venn diagram.

- - -- ---

This is a 4-set venn diagramm btw:

--- -- - -

So, back to me attempting to do the 4th dedekind-number visualization:

So I started to draw that venn diagram using ellipses, but I slightly altered it, because I wanted to use my isometric grid paper. (The angles of my drawing are different than in the picture depicted above.)

I started to draw the 4 ellipses, and I somehow started to see two intertwined/interlocked tubes due to the additional helplines I used for drawing.

Then I started to use my thicker black pen to make this effect of these two interlocked tubes more visible:

Now I plan to write the sets/letters at the backside of that piece of paper - in mirrored, so I can hold this piece of paper (with the tube drawing in front) against the light - and see the letters of the set names shining through.

i might also add some details to the tubes.

#my train of thought is the most dispersed scattered mess of whatever one might call this fuckery #... and yet in the end this clusterfuck of incoherent thought ramblings still make sense #the nonsense facility works nonsensically at producing non-nonsensical side-products #clusterfuck of nonsense thought ramblings #math #math or such idk #math art #dedekind numbers #venn diagram #venn diagrams #4th dedekind number #hypercube #set theory #sets #optical illusion #artsy #art #mathy art #my art #wip