Timeline for Which platonic solids can form a topological torus?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Feb 4, 2011 at 1:28 | comment | added | Tracy Hall | It's not on this page, but found in the top answer to math.stackexchange.com/questions/19386/tetrahedral-torus that Willie Wong linked to in a comment to the question. | |
| Feb 1, 2011 at 2:20 | comment | added | sleepless in beantown | @Tracy-Hall, my eyes must be missing the reference to the 1972 paper on this page. What/where exactly is the paper you're referring to? Thanks for the Martin Garner reference. | |
| Jan 29, 2011 at 7:14 | vote | accept | fastforward | ||
| Jan 29, 2011 at 2:56 | comment | added | Tracy Hall | The same answer is also referenced on page 150 of Martin Gardner's 1987 "Time Travel and Other Mathematical Bewilderments". He credits Kurt Schmucker for the 8-polyhedron solutions for the four larger platonic solids and the same paper of J. H. Mason for the impossibility proof for tetrahedra. | |
| Jan 28, 2011 at 23:54 | comment | added | Tracy Hall | Actually, there's no need to keep track of the numerators mod 3, and the same proof works for regular simplices of any dimension $d$, by noticing which entry of the multiply-reflected all-ones vector has the lowest power in the denominator of $d$, in odd dimensions, or $\frac d2$, in even dimensions greater than $2$. It follows that the representation of the Coxeter group (whose graph is a complete graph with all edges labelled by infinity) is faithful for all $d>2$, and that no such solid torus loop exists for any higher-dimensional regular simplex. I think that finishes the question. | |
| Jan 28, 2011 at 22:41 | history | edited | Tracy Hall | CC BY-SA 2.5 |
Improve twelve to eight; general case for opposite facets.
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| Jan 28, 2011 at 22:12 | history | answered | Tracy Hall | CC BY-SA 2.5 |