apl. Prof. Dr. Sascha Kurz

Lehrstuhl für Wirtschaftsmathematik
Mathematisches Institut, Fakultät für Mathematik, Physik und Informatik

Heronian Figures

Integral point sets

Through the ages mathematicians were interested in geometric figures with integral side lengths, areas and volumes. We call a set of n points in the d-dimensional Euclidean space heronian if all pairwice distances are integral and the k-dimnesional volume of each k+1 subset of the n points is rational for k<=d. In many cases rational implies integral.

Primitive Heronian Triangles per diameter

Number of primitive Heronian Triangles with diameter at most n

n # log2(#/n) / log2(log2(n)) log2(#)/log2(n)
1 0    
2 0    
4 0    
8 3    
9 3    
16 7    
32 26    
64 78 0,110408649557828 1,04756703647704
99 166 0,267634576338556 1,11005404402003
128 266 0,375899187954382 1,15075463364303
256 752 0,518196283892546 1,1943236064597
512 2291 0,681959901991711 1,24019574925608
999 6073 0,784571129077055 1,26113442704085
1024 6332 0,791240949550506 1,26284455401372
2048 17415 0,892644570812616 1,28073116840674
4096 47313 0,984654377920019 1,29416241841785
8192 126061 1,06575509591822 1,30336634513439
9999 166686 1,08763085734359 1,30547478121364
16384 331121 1,13911076690033 1,30978564179477
32768 860078 1,20661376663631 1,31427386516021
65536 2202824 1,2677307003315 1,31693267508287
99999 3889418 1,30279943420061 1,31797692395159
131072 5586001 1,32438742845082 1,31843437654187
262144 14018584 1,37672435213453 1,31893540533668
524288 34867248 1,42548787490153 1,31870364020392
600000 41589538 1,43458757825675 1,31858509164126
999999 80872450 1,4680567202640 1,3179667666826

At least for small n, we have #≤ n*(log2(n))2. A more detailed list can be found here. Thanks to Randall Rathbun for making available the data for large diameters.

Number of primitive Heronian tetrahedrons with diameter at most n

n # remark
64 0  
128 1  
256 4  
512 9  
1024 27  
2048 62  
4096 129  
8192 218  
16384 394  
32768 620  
60000 824 coincides with Jan Fricke's count
65536 865  
131072 1263  
262144 1688  
524288 2369  
600000 2529  

A complete list of all primitive Heronian tetrahedrons with diameter at most 600000 (and other examples) can be obtained here.

Primitive Heronian k-gons a.k.a. k_2-cluster

More information of n_m-clusters can be found at the homepage of Landon Curt Noll.

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Letztes Update am: 3.08.2017

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