#math model

[art-of-mathematics]

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. )

I will describe this model and the math of it once I have finished it.

[art-of-mathematics]

Wobbly Sierpinski-fractal model/toy

Made with foil strips of equal length and book screws.

The larger triangle at the center is stable because the outer tinier triangles stabilize it. (The larger triangles have a side length of two foil strips. The smallest triangles ("the building blocks") have a side length of 1 foil strip. The largest triangle is this entire model with side length of 4 foil strips.)

The largest triangle (4foil strips) is stable, because the tiniest (1foil strip) triangles stabilize each other as well as the larger (2foil strips) triangle.

The sierpinski pattern has interesting properties regarding its "stability" I have to admit.

Normally I struggle to make equilateral triangles of side length "2 foil strips" stable enough without adding an inner triangle. This construction (using outer tiny triangles) is a good solution if you want a "larger triangle hole".

[art-of-mathematics]

Hyper-tetrahedron Tetrahedral prism model

That is a model I made back in ~2013 with paper clips.

Back then I attempted to build tiny chain reactions (marble tracks) at the wall of my room. I remember building a cube-like marble releaser using only sticks and tie. - Cubes are not stable like tetrahedra when only using sticks for the main edges. Without additional edges the cube could be distorted by torsion. This made it annoying as it resulted in needing far more sticks to stabilize. -> Tetrahedra are more efficient in these aspects.

- - -- --- -----

The molecular structure of diamonds is also tetrahedral. (It could also explain why diamonds are so hard/stable. )

- - - (correction after posting) - - -

hyper-tetrahedron/5-cell/4-simplex (Left) and tetrahedral prism (Right):

[art-of-mathematics]

Self-similar tetrahedra packing with 4 smaller tetrahedra and 1 octahedron

That was my contemplation some months ago:

Today I started a trashy/bad-quality prototype 3D model with paper - and it appears like a first step for a proof to my contemplation:

- 4 tetrahedra with side length of 3 cm

- 1 tetrahedron with side length of 6 cm

- 1 octahedron with side length of 3cm

-> the 4 small tetrahedra and the octahedron packed together fill the large tetrahedron - wonderfully packed 3D space. There are no gaps. (A better quality model might depict it better than this trashy model. )

Next step: I will craft a better quality 3D model.

[art-of-mathematics]

Icosahedron

Today I improved the thread connections.

[art-of-mathematics]

I wanted to carry the icosahedron model - but didn"t know how to carry it without causing damage - as my backpack is pretty stuffed.

I quickly grabbed 12 tiny strips of foil with holes in their ends, 6 screwable rivets, and a lanyard - and built an octahedron net I can carry around my neck.

The chance I might lose the icosadron model/ "inlay" is a bit higher as with my other plastic foil and rivet constructions - as - if I lose one rivet the hole in this net is large enough for the icosahedron inlay to fall through. (The rivets are just screwed in - so I can also take the inlay icosahedron model out. ). And losing one rivet is very probable.

But I take the risk - as I would most certainly notice the colorful icosahedron falling down if I lose a rivet. (The tiny rivet would most certainly be lost...But that loss is rather unimportant.)

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Side thoughts:

I love how such strips-and-rivets models are helpful for visualizing graph networks.

The rivets are nodes.

The strips are edges.

-> Also:

[Sorry for the incoming word salad/ thought gibberish:]

The stability of the physical models is better the more connections(/strips/edges) exist between the rivets/nodes. (More connections per rivet and more stability go hand in hand - literally. Losing one rivet/node would have less impact on the overall stability the more connected all rivets/nodes are towards each other.)

[ Further thoughts: Similar principles exist in the memory and how more connected memories are less prone to be forgotten. This is often the case for the learning technic of "Eselsbrücken" - i dont know the english equivalent rn- it means using associations to memorize stuff more easily - so it sticks better in the memory. (I might add an example later. But am too lazy rn)

Memory works like a dynamical graph network... ]

[art-of-mathematics]

A tiny klein bottle with a tiny flourescent stand

The klein bottle is from Cliff Stoll (ACME klein bottles). The green bottom piece is a tiny flourescent silicone container. The bronze-colored wire was a safety pin. The band around the green container is a blue-and-green hair tie.

The flourescence of the green container/bottom is relatively bright.

[art-of-mathematics]

I crafted this tiny tetrahedron key chain from a triangle rest of the PVC foil I used for another model some days ago. I could not throw the tiny triangle in the trash can.... Better fold it into 4 tiny triangles - and create a tiny tetrahedron.

Yeah, and i added those 2 tiny triskelion beads for aesthetic reasons.