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 tT12184p3+4p6 t{3,3}
tC24366p8+8p3 t{4,3}
tO24366p4+8p6 t{3,4}
tD609020p3+12p10 t{5,3}
tI609020p6+12p5 t{3,5}
TT = O612(4+4)p3 r{3,3} = {31,1}011
CO12246p4+8p3 r{3,4}
ID306020p3+12p5 r{3,5}
Pq2q4q2pq+qp4 t{2,q}
APq2q4q2pq+2qp3 s{2,2q}
semiregular 4-polytopes
4 tC51030(10+20)p35O+5T r{3,3,3} = {32,1}021
tC8329664p3+24p48CO+16T r{4,3,3}
tC16=C24(*)489696p3(16+8)O r{3,3,4}
tC249628896p3 + 144p424CO + 24C r{3,4,3}
tC6007203600(1200 + 2400)p3600O + 120I r{3,3,5}
tC120120036002400p3 + 720p5120ID+600T r{5,3,3}
HM4 = C16(*)82432p3(8+8)T {3,31,1}111
–306020p3 + 20p6(5 + 5)tT 2t{3,3,3}
–288576192p3 + 144p8(24 + 24)tC 2t{3,4,3}
–206040p3 + 30p410T + 20P3 t0,3{3,3,3}
–144576384p3 + 288p448O + 192P3 t0,3{3,4,3}
–q22q2q2p4 + 2qpq(q + q)Pq 2t{q,2,q}
semiregular 5-polytopes
5 S511560(20+60)p330T+15O6C5+6tC5 r{3,3,3,3} = {33,1}031
S522090120p330T+30O(6+6)C5 2r{3,3,3,3} = {32,2}022
HM51680160p3(80+40)T16C5+10C16 {3,32,1}121
Cr5140240(80+320)p3160T+80O32tC5+10C16 r{3,3,3,4}
Cr5280480(320+320)p380T+200O32tC5+10C24 2r{3,3,3,4}
semiregular 6-polytopes
6 S61 (*) r{35} = {34,1}041
S62 (*) 2r{35} = {33,2}032
HM632240640p3(160+480)T32S5+12HM5 {3,33,1}131
V2727216720p31080T72S5+27HM5 {3,3,32,1}221
V72727202160p32160T(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(*)646722240p3(560+2240)T64S6+14HM6 {3,34,1}141
V56567564032p310080T576S6+126Cr6 {3,3,3,32,1}321
V126126201610080p320160T576S6+56V27 {3,3,33,1}231
V5765761008040320p3(30240+20160)T126HM6+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(*)12817927168p3(1792+8960)T128S7+16HM7 {3,35,1}151
V2160216069120483840p31209600T17280S7+240V126 {3,3,34,1}241
V240240672060480p3241920T17280S7+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:

  • Vn = semiregular polytope with n vertices

Polygons

Polyhedra:

4-polytopes:

See also

Notes

  1. 1 2 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
  3. ↑ Coxeter, H.S.M. Regular polytopes, 3rd Edn, Dover (1973) p. 210 (11.x Historical remarks)
  4. ↑ Page 128
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