A272388 Longest side of Heronian tetrahedron.

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

Offset: 1

Author

Albert Lau, May 19 2016

Keywords

• nonn

Comments

A Heronian tetrahedron or perfect tetrahedron is a tetrahedron whose edge lengths, face areas and volume are all integers.

Examples

			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.
		

Links

• R. H. Buchholz, Perfect Pyramids, Bull. Austral. Math. Soc. 45, 353-368, 1992.
• Susan H. Marshall and Alexander R. Perlis, Heronian Tetrahedra Are Lattice Tetrahedra, American Mathematical Monthly 120:2 (2013), 140-149.
• Ivars Peterson, Perfect pyramids.
• Eric Weisstein's World of Mathematics, Heronian Tetrahedron.

Crossrefs

Cf. A120131, A120132, A120133.

Programs

Mathematica

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

Extensions

a(11)-a(53) from Giovanni Resta, May 20 2016