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# Math formula(e) needed: tesseract model construction? Answered

Although I CAN do this without the math formula(e) to figure it all out, I would like to have it for a quicker method of figuring out size, etc. so I didn't have to use up so much material in guessing.

I would like to construct the tesseract I have pictured here, using rigid clear plastic sheets or panels.

Given a said size for the "inner cube" say 2 inches (approx. 5 cm or about 51 mm) what formula(e) must I use to calculate the panels' dimensions to connect to the larger cube (knowing full well this will depend on the size of the larger cube....say approx. 4 inches across (about 10 cm or 102 mm)?

The reason I'd like the formula(e) is that I may need to increase the size of the center cube by an inch or so, and this would allow me to do so without having to go through the "trial and error" thing so many times.

I would, of course, want the center cube to be "centered", if possible or shall I say as close as possible.

Thanks....

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## Discussions

For all the math and geometry folks out there, this problem demonstrates very nicely the generalization of Pythagoras' Theorem to N dimensions: The body diagonal H of an N-dimensional rectangle is given by

H2 = SUM si2

where s_i is the length of the ith side. For the case of an N-cube with side s, this reduces to s*sqrt(N).

For your nested cubes, Goodhart, with sides L and M (L > M), each of your red lines is just half the difference of the two body diagonals:

[sqrt(3)L - sqrt(3)M] / 2 = (L-M) sqrt(3)/2

which is BritLuv's solution.

Trigonometry and pythagoras' theory would be helpful. If you work out the distance from the corner to the point equally as high as the inner corner, and then between the outer line to the inner line, you can work out the hypotenuse (longest side) because there should be a right angle between the two distances you've worked out. If we name these two distances 'a' and 'b', and the hypotenuse 'c':
a(squared) + b(squared) = c(squared)
All you need to do is work out the square root of c and you have the length of the red lines.
Providing the cube in the middle is perfectly central, all of the red lines would be the same.
Hope this helps.

For me, algebra becomes confusing because, when I asked people to "show me another way" to get the the same answer (for checking purposes), I never could get any answers. With geometry, I could figure things out with the minimum of information....it is very logical for us lateral thinkers :-)

Acrylic laser cuts very very nicely. If you want the surfaces getting ready for laser cutting I don't mind doing it for you. It probably wouldn't be economical for me to laser cut them then send them to you, though I can if you really like.

If I were doing it I'd cut the surfaces from 3mm acrylic and hold them together with strips of OHP transparency and double sided tape. I think....

What's it for? Art?

. As you know, I'm no Mathematician, so just consider this a let's-see-how-close-I-can-get guess.
. It seems to me that if you extend a red line to the center of the inside cube, it becomes the hypotenuse of two right triangles - one part of the inside cube and the other (?colinear? with the first) is part of the outside cube. You know the dimensions of each edge, use Trig to calc the rest.
. Waited with bated breath to see if I'm even in the ballpark.

This should work... basically, usually the a^2 + b^2 = c^2 method:

outside square [length/2^2 + length/2^2 = length of red line to center of cubes^2]
MINUS
inside square [length/2^2 + length/2^2 = length of red line to center of cubes^2]

so... if your outside cube is 4x4x4 and the inside is 2x2x2, that should make the length of the red lines sqrt8 - sqrt2. Roughly 1.42 inches.

So i drew it out and measured the red line... it comes out to 1 and 7/16... seemingly confirming my answer. The green line is the wrong hypotenuse, I'm figuring it to the middle of the cube, not off an outside square. Granted, it's been a long time and I could be wrong, my formula is probably not as simple as it could be, but it _should_ be correct. Interestingly, if you use 2 instead of 3 as the square root in your formula, it comes out to be the same as my answer.