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May
14
comment How to determine whether a polytope is self-tessellating?
As for the non-convex part, I didn't know about the Soma cube. Interesting, in particular the last bit! What software did you use to create the illustrations?
May
14
comment How to determine whether a polytope is self-tessellating?
Thanks, great explanation! Couple of short questions. In the first part, what does it mean for vertices to be similar? Same valency? Second part, this does not disprove the existence of other convex self-tessellating polytopes (i.e. non-prisms), right? Finally, could you perhaps recommend some comprehensible literature on polytopes in general (other than Coxeter's works)?
May
14
awarded  Benefactor
May
14
accepted How to determine whether a polytope is self-tessellating?
May
9
comment Prove or disprove that EigenVectors of $A^T$ are the same of $A$.
Would it help to mention that $A$ and $A^T$ share their eigenvalues? In other words, we have $A v_1 = \lambda_1 v_1$ and $A^T w_1 = \lambda_1 w_1$. If I understand your question correctly, you now have to prove that $v_1$ and $w_1$ are not always parallel.
May
9
awarded  Organizer
May
8
comment Extrema of quartic functions
It might help to consider the derivative $\alpha x^3 + \beta x^2 + \gamma x + \delta$, where $\alpha = 4a$, $\beta = 3b$, $\gamma = 2c$ and $\delta = d$. Then, setting e.g. $\alpha=1$, compute $\Delta = 18\beta\gamma\delta - 4\beta^3\delta + \beta^2\gamma^2 - 4\gamma^3 - 27\delta^2$. If $\Delta > 0$ there are three distinct roots (see this article) which would indicate two local minima and your local maximum. Not sure yet how to continue from here.
May
7
comment How to determine whether a polytope is self-tessellating?
@DavidK Thanks. Hmm, the book certainly explains the concept of honeycombs but does not seem to discuss the notion of self-tessellation (note — I'm not aware of a proper existing mathematical term that describes the concept). The condition of equal-sized parts is a reasonable first step, I've updated the question.
May
7
revised How to determine whether a polytope is self-tessellating?
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May
7
revised How to determine whether a polytope is self-tessellating?
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May
7
revised How to determine whether a polytope is self-tessellating?
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May
7
comment How to determine whether a polytope is self-tessellating?
@pjs36 Thanks for the link, contains some interesting information and references :). As for the question, I'm not looking for polytopes that can tessellate space, but rather polytopes that can tessellate themselves. The former does not imply the latter, take for example a hexagon. It can be used to tile $\mathbb{R}^2$ but a hexagon cannot be tiled using a finite number of scaled hexagons. The latter might imply the former though.
May
7
revised How to determine whether a polytope is self-tessellating?
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May
7
revised How to determine whether a polytope is self-tessellating?
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May
6
revised How to determine whether a polytope is self-tessellating?
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May
6
revised How to determine whether a polytope is self-tessellating?
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May
6
comment How to determine whether a polytope is self-tessellating?
@pjs36 Sure, as long as they're self-tessellating. All examples on that page are zonotopes by the way, i.e. projections of hypercubes (hence the nice properties).
May
6
revised How to determine whether a polytope is self-tessellating?
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May
6
asked How to determine whether a polytope is self-tessellating?
Apr
30
awarded  Notable Question