Among the first dozen, only 5,7,8 has a rational angle. Can you rule out others? Or at least pythagoreans? --rwg On 2015-08-12 07:43, Ed Pegg Jr wrote:
8 copies of any triangle can make a convex 4-scalenohedron. Some triangles can make 3 distinct convex 4-scalenohedra. If integer triangles are used, some of the convex 4-scalenohedra have rational volume. I haven't yet found a triangle that admits two integer volumes. So far, all non-integral volumes have a denominator of 3.
Here are some solutions I've found so far, with the triangle sides followed by the volume.
05,07,08 -- 96 07,13,14 -- 384 09,21,22 -- 3200/3 15,18,23 -- 4928/3 14,19,25 -- 2112 16,19,23 -- 2400 11,31,32 -- 2400 19,21,28 -- 2400 17,20,23 -- 3360 13,43,44 -- 4704 22,25,27 -- 6720 15,57,58 -- 25088/3 25,27,32 -- 8736 21,37,42 -- 31360/3 24,39,43 -- 39200/3 17,73,74 -- 13824 28,35,37 -- 15840 27,38,39 -- 49280/3 31,34,37 -- 17472 37,40,53 -- 18816 33,37,46 -- 18816 42,43,59 -- 19584 28,41,47 -- 20064 19,91,92 -- 21600 35,39,56 -- 24480 41,47,62 -- 26880 35,42,43 -- 28224 38,49,61 -- 31680 37,43,46 -- 32256 38,43,55 -- 34944 39,49,58 -- 38400 44,49,63 -- 38688 41,46,63 -- 42240 32,67,71 -- 42336 37,58,61 -- 47424 43,52,61 -- 50400 45,53,58 -- 59136 43,54,67 -- 59136 41,59,70 -- 61248 49,53,54 -- 65280 49,63,92 -- 69600 49,59,68 -- 76032 55,68,87 -- 82656 61,65,86 -- 86400 56,61,77 -- 86400 51,62,63 -- 267520/3 50,67,87 -- 94080 47,71,76 -- 97440 47,80,83 -- 110880 51,79,88 -- 117600 61,63,76 -- 119040 59,67,74 -- 126720 59,67,76 -- 131040 64,65,71 -- 137088 51,86,87 -- 412160/3 59,77,86 -- 149760 65,66,79 -- 150528 71,73,76 -- 184800 69,79,92 -- 198912 67,80,87 -- 205920
I've also found various bipyramids and other 8-triangle shapes with rational volumes.
03,03,04 -- 32/3 08,09,09 -- 896/3 17,17,24 -- 384 11,11,12 -- 672 12,19,19 -- 1632 16,33,33 -- 15872/3 20,27,27 -- 18400/3 20,31,31 -- 7392 20,51,51 -- 39200/3 43,43,60 -- 16800 33,33,40 -- 54400/3 28,51,51 -- 73696/3 24,73,73 -- 27264 57,57,80 -- 89600/3 41,41,48 -- 35328 36,83,83 -- 68256 56,57,57 -- 257152/3 59,59,60 -- 98400 67,67,84 -- 145824 72,89,89 -- 252288
There's probably something of interest in these somewhere.
--Ed Pegg Jr _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun