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 Apr 8 awarded Notable Question Feb 24 awarded Popular Question Jan 10 revised Constructing a spherical triangle of a given surface area added 95 characters in body Jan 10 revised Can even degree Legendre polynomials have roots in common? added 105 characters in body Jan 7 revised Can even degree Legendre polynomials have roots in common? added 36 characters in body Jan 7 revised Constructing a spherical triangle of a given surface area deleted 26 characters in body Jan 6 revised Constructing a spherical triangle of a given surface area added 1 character in body; edited title Jan 6 revised Constructing a spherical triangle of a given surface area edited title Jan 6 revised Constructing a spherical triangle of a given surface area edited body Jan 6 revised Constructing a spherical triangle of a given surface area added 88 characters in body Jan 6 asked Constructing a spherical triangle of a given surface area Sep 27 awarded Popular Question 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.