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A272388
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Longest side of Heronian tetrahedron.
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1
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117, 160, 203, 225, 234, 318, 319, 319, 320, 351, 406, 429, 450, 468, 468, 480, 585, 595, 595, 595, 609, 612, 636, 638, 638, 640, 671, 675, 680, 680, 697, 697, 702, 741, 780, 800, 812, 819, 858, 884, 884, 888, 900, 925, 935, 936, 936, 954, 957, 957, 960, 990, 990
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OFFSET
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1,1
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COMMENTS
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A Heronian tetrahedron or perfect tetrahedron is a tetrahedron whose edge lengths, face areas and volume are all integers.
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LINKS
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EXAMPLE
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The following are examples of Heronian tetrahedra.
dAB, dAC, dBC, dCD, dBD, dAD, SABC, SABD, SACD, SBCD, Volume
117, 84, 51, 52, 53, 80, 1890, 1800, 2016, 1170, 18144
160, 153, 25, 39, 56, 120, 1872, 2688, 1404, 420, 8064
203, 195, 148, 203, 195, 148, 13650, 13650, 13650, 13650, 611520
225, 200, 65, 119, 156, 87, 6300, 4914, 2436, 3570, 35280
234, 168, 102, 104, 106, 160, 7560, 7200, 8064, 4680, 145152
318, 221, 203, 42, 175, 221, 22260, 18564, 4620, 2940, 206976
319, 318, 175, 175, 210, 221, 26796, 23100, 18564, 14700, 1034880
319, 318, 175, 203, 252, 221, 26796, 27720, 22260, 17640, 1241856
320, 306, 50, 78, 112, 240, 7488, 10752, 5616, 1680, 64512
351, 252, 153, 156, 159, 240, 17010, 16200, 18144, 10530, 489888
where
dPQ is the distance between vertices P and Q and
SPQR is the area of triangle PQR.
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MATHEMATICA
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aMax=360(*WARNING:takes a long time*);
heron=1/4Sqrt[(#1+#2+#3)(-#1+#2+#3)(#1-#2+#3)(#1+#2-#3)]&;
cayley=1/24Sqrt[2Det[{
{0, 1, 1, 1, 1},
{1, 0, #1^2, #2^2, #6^2},
{1, #1^2, 0, #3^2, #5^2},
{1, #2^2, #3^2, 0, #4^2},
{1, #6^2, #5^2, #4^2, 0}
}]]&;
Do[
S1=heron[a, b, c];
If[S1//IntegerQ//Not, Continue[]];
Do[
S2=heron[a, e, f];
If[S2//IntegerQ//Not, Continue[]];
Do[
If[b==e&&c>f||b==f&&c>e, Continue[]];
S3=heron[b, d, f];
If[S3//IntegerQ//Not, Continue[]];
S4=heron[c, d, e];
If[S4//IntegerQ//Not, Continue[]];
V=cayley[a, b, c, d, e, f];
If[V//IntegerQ//Not, Continue[]];
If[V==0, Continue[]];
a//Sow(*{a, b, c, d, e, f, S1, S2, S3, S4, V}//Sow*);
, {d, Sqrt[((b^2-c^2+e^2-f^2)/(2a))^2+4((S1-S2)/a)^2]//Ceiling, Min[a, Sqrt[((b^2-c^2+e^2-f^2)/(2a))^2+4((S1+S2)/a)^2]]}];
, {e, a-b+1, b}, {f, a-e+1, b}];
, {a, 117, aMax}, {b, a/2//Ceiling, a}, {c, a-b+1, b}]//Reap//Last//Last
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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