Real World Tesseract: Crafting a 4D Table from 3D Cubes

[DaveC426913]

It is with mixed success that I finish my third attempt at what I hope is the world's first real-world (i.e. not computer-rendered) tesseract, or hypercube.

It started last spring while I was bored in a job-hunting class. I was studying my Meyer-Briggs results, and the chart that divides all people into one of 16 groups. I wanted a chart that could represent all 16 types in one symmetrical arrangement

If there were only two properties, all combinations could be represented in a simple table:

Introverted/Extroverted
Sensing/iNtuiting

           [B]I[/B](1stcolumn)  [B]E[/B](2ndcolumn)
[B]S[/B](1strow)   IS         ES
[B]N[/B](2ndrow)   IN         EN

With three properties, you'd need a cubic table

Introverted/Extraverted
Sensing/Intuiting
Thinking/Feeling

Thinking         Feeling
    [B]I[/B]     [B]E[/B]        [B]I[/B]     [B]E
S[/B]  IS[B]T[/B]   ES[B]T[/B]      IS[B]F[/B]   ES[B]F[/B]                 
[B]N[/B]  IN[B]T[/B]   EN[B]T[/B]      IN[B]F[/B]   EN[B]F[/B]

On this webpage, I'm forced to flatten it to 2 dimensions, and have the third stick out the side. This produces an unintentional bias toward one property over the others. In the above example, Sensing/Intuiting is singled out as two tables.

But as a cubic table, I can represent all three properties in their correct relationship to each other. All I have to do is flip the cube so I'm looking at the properties I want.
Property 1: columns
Property 2: rows
Property 3: depthsBut ... with four properties, you can't represent them all at the same time in correct relation to each other. Jjust like stacking two 2D tables got me a 3D cubic table, I'd like to be able to stack two 3D cubic tables to make one 4D table

Introverted/Extroverted
Thinking/Feeling
Sensing/Intuiting
Judging/Perceiving

Judging

[B]T[/B](frontlayer)       [B]F[/B](rearlayer)
   [B] I      E         I      E
S[/B]  IST[B]J[/B]   EST[B]J[/B]      ISF[B]J[/B]   ESF[B]J
N[/B]  INT[B]J[/B]   ENT[B]J[/B]      INF[B]J[/B]   ENF[B]J[/B]Perceiving

[B]T[/B](frontlayer)       [B]F[/B](rearlayer)
[B]    I      E         I      E
S[/B]  IST[B]P[/B]   EST[B]P[/B]      ISF[B]P[/B]   ESF[B]P
N[/B]  INT[B]P[/B]   ENT[B]P[/B]      INF[B]P[/B]   ENF[B]P[/B]

So, I need a four dimensional table - all four properties in the correct relation to each other. A tesseract nicely fits this bill.

The fourth property is represented by a fourth dimnesion in the table.
Property 1: columns
Property 2: rows
Property 3: depths
Property 4: ?
Now, since the physical world can't actually represent a 4th dimension, it gets squashed. This squashed layer can be represented by a 4th squahsed "direction": layers. Inner and outer.

Thus begins my quest for a real-world tesseract.

 

[DaveC426913]

The Mark I

My first attempt is with chemistry building blocks. The vertices are all carbon atoms.

I didn't work out because the lengths of the sides have to change. The "struts" that join the inner and outer cubic tables are shorter. But this model doesn't do that; the structure doesn't collapse properly; it won't easily turn inside out.

 

[DaveC426913]

The Mark II

This one is built with great pains (trust me on this) to make contractable/expandable struts like pistons (32 struts, each made of two pistons for a total of 64).

This had a very interesting lesson: when we allow the device to do what it needs to do to rotate, it now does it too easily. All 32 struts must change length at the same time. This model would not hold its shape long enough to even get it looking like a tesseract, it just collapsed like so many sticks in a game of pickup sticks. I couldn't even set it up to take the pic!

You can see the outer cube, the the inner one just falls flat.

I need a model that is self-supporting.

 

[dst]

I was going to suggest magnetic sticks with ball bearing vertices but it doesn't look like it's working,

 

[DaveC426913]

The Mark III

My next attempt, which I've just finished, is made of even more permanent material, and more exacting measurements and its solid structures provide some support so that it does not completely collapse in on itself.

The basic subunit is a cube with four corners truncated and a joint in the face of each. (The cubes, all 16 of them, with their 4 truncated vertices apiece, have been cut and sanded to within less than 1mm) The subunit cubes are joined by stretchy elastic cord.

Interestingly, while a a cube has three edges to every vertex, a hypercube has 4 edges to every vertex. One side-effect of this is that every subunit has an even number of cords attached to it. That means no subunit has an end of the cord, the yare all "pass-through". That entire elastic cord is one single piece, threaded through out the entire model. In fact, it has zero ends - it's a loop.

That aside though, I'm still having some troubles. It still doesn't easily transform from one shape to another. Too much friction and too much loosey-goosey.

I have yet to create one that smoothly transofmrs from one shape into the next without a tremendous amount of awkward, show-stopping fiddling.

Also, I have not yet managed to get the thing to open a portal to an alternate dimension, making me rich and famous beyond my wildest dreams of avarice.


I continue on my quest.

 

[Gokul43201]

I would have suggested hollow, low-friction, telescoping rods with an internal, low-stiffness spring.

 

[DaveC426913]

I would have suggested hollow, low-friction, telescoping rods with an internal, low-stiffness spring.

That is how the Mark II is built.

The low-friction thing was what I worked very carefully on. That was the rationale for the hours of precision work. Strangely, it turned out to be the wrong thing. Those pistons have zero frction. So little that I can't even get the thing to stay put; I basically have to support every single strut at all times or it collapses into what you see. (And I don't mean standing on its own, I mean when I'm supporting it in my hands, I can only hold a dozen or so struts. Any that I'm not holding just flop like an old woman's funbags.) That's what all that tape is on there for in the pic above; it's there to add friction.

As for the spring, that's pretty tricky. Sometimes it needs to have compression tension, soemtimes it needs expansion tension. That was the rationale ofr the Mark III. The solid shape provides an automatic spacer - two forces in opposite directions (so that the spring can always be compression.)

 

[Gokul43201]

I was thinking that the spring would be good because it has a natural length that it'll want to retain. That would prevent the kind of collapse that happened with MkII. Also, some friction is good, as a damping force. Ideally, you'd want the system to be optimally damped.