Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibór)[1] was a Dutch mathematician. He is noted for discovering and classifying semiregular polytopes in dimensions four and higher.

Elte's father Hartog Elte was headmaster of a school in Amsterdam. Emanuel Elte married Rebecca Stork in 1912 in Amsterdam, when he was a teacher at a high school in that city. By 1943 the family lived in Haarlem. When on January 30 of that year a German officer was shot in that town, in reprisal a hundred inhabitants of Haarlem were transported to the Camp Vught, including Elte and his family. As Jews, he and his wife were further deported to Sobibór, where they were murdered; his two children were murdered at Auschwitz.[1]

Elte's semiregular polytopes of the first kind

His work rediscovered the finite semiregular polytopes of Thorold Gosset, and further allowing not only regular facets, but recursively also allowing one or two semiregular ones. These were enumerated in his 1912 book, The Semiregular Polytopes of the Hyperspaces.[2] He called them semiregular polytopes of the first kind, limiting his search to one or two types of regular or semiregular k-faces. These polytopes and more were rediscovered again by Coxeter, and renamed as a part of a larger class of uniform polytopes.[3] In the process he discovered all the main representatives of the exceptional En family of polytopes, save only 142 which did not satisfy his definition of semiregularity.

Summary of the semiregular polytopes of the first kind[4]
n Elte
notation 		Vertices Edges Faces Cells Facets Schläfli
symbol 		Coxeter
symbol 		Coxeter
diagram
Polyhedra (Archimedean solids)
3 tT 12 18 4p3+4p6 t{3,3}
tC 24 36 6p8+8p3 t{4,3}
tO 24 36 6p4+8p6 t{3,4}
tD 60 90 20p3+12p10 t{5,3}
tI 60 90 20p6+12p5 t{3,5}
TT = O 6 12 (4+4)p3 r{3,3} = {31,1} 011
CO 12 24 6p4+8p3 r{3,4}
ID 30 60 20p3+12p5 r{3,5}
Pq 2q 4q 2pq+qp4 t{2,q}
APq 2q 4q 2pq+2qp3 s{2,2q}
semiregular 4-polytopes
4 tC5 10 30 (10+20)p3 5O+5T r{3,3,3} = {32,1} 021
tC8 32 96 64p3+24p4 8CO+16T r{4,3,3}
tC16=C24(*) 48 96 96p3 (16+8)O r{3,3,4}
tC24 96 288 96p3 + 144p4 24CO + 24C r{3,4,3}
tC600 720 3600 (1200 + 2400)p3 600O + 120I r{3,3,5}
tC120 1200 3600 2400p3 + 720p5 120ID+600T r{5,3,3}
HM4 = C16(*) 8 24 32p3 (8+8)T {3,31,1} 111
– 30 60 20p3 + 20p6 (5 + 5)tT 2t{3,3,3}
– 288 576 192p3 + 144p8 (24 + 24)tC 2t{3,4,3}
– 20 60 40p3 + 30p4 10T + 20P3 t0,3{3,3,3}
– 144 576 384p3 + 288p4 48O + 192P3 t0,3{3,4,3}
– q2 2q2 q2p4 + 2qpq (q + q)Pq 2t{q,2,q}
semiregular 5-polytopes
5 S51 15 60 (20+60)p3 30T+15O 6C5+6tC5 r{3,3,3,3} = {33,1} 031
S52 20 90 120p3 30T+30O (6+6)C5 2r{3,3,3,3} = {32,2} 022
HM5 16 80 160p3 (80+40)T 16C5+10C16 {3,32,1} 121
Cr51 40 240 (80+320)p3 160T+80O 32tC5+10C16 r{3,3,3,4}
Cr52 80 480 (320+320)p3 80T+200O 32tC5+10C24 2r{3,3,3,4}
semiregular 6-polytopes
6 S61 (*) r{35} = {34,1} 041
S62 (*) 2r{35} = {33,2} 032
HM6 32 240 640p3 (160+480)T 32S5+12HM5 {3,33,1} 131
V27 27 216 720p3 1080T 72S5+27HM5 {3,3,32,1} 221
V72 72 720 2160p3 2160T (27+27)HM6 {3,32,2} 122
semiregular 7-polytopes
7 S71 (*) r{36} = {35,1} 051
S72 (*) 2r{36} = {34,2} 042
S73 (*) 3r{36} = {33,3} 033
HM7(*) 64 672 2240p3 (560+2240)T 64S6+14HM6 {3,34,1} 141
V56 56 756 4032p3 10080T 576S6+126Cr6 {3,3,3,32,1} 321
V126 126 2016 10080p3 20160T 576S6+56V27 {3,3,33,1} 231
V576 576 10080 40320p3 (30240+20160)T 126HM6+56V72 {3,33,2} 132
semiregular 8-polytopes
8 S81 (*) r{37} = {36,1} 061
S82 (*) 2r{37} = {35,2} 052
S83 (*) 3r{37} = {34,3} 043
HM8(*) 128 1792 7168p3 (1792+8960)T 128S7+16HM7 {3,35,1} 151
V2160 2160 69120 483840p3 1209600T 17280S7+240V126 {3,3,34,1} 241
V240 240 6720 60480p3 241920T 17280S7+2160Cr7 {3,3,3,3,32,1} 421
(*) Added in this table as a sequence Elte recognized but did not enumerate explicitly

Regular dimensional families:

Semiregular polytopes of first order:

Polygons

Polyhedra:

4-polytopes:

See also

Notes

  1. ^ a b Emanuël Lodewijk Elte at joodsmonument.nl
  2. ^ Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen, ISBN 1-4181-7968-X [1] [2]
  3. ^ Coxeter, H.S.M. Regular polytopes, 3rd Edn, Dover (1973) p. 210 (11.x Historical remarks)
  4. ^ Page 128
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