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Emanuel Lodewijk Elte (16 March 1881, Amsterdam - 9 April 1943, Sobibor)[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 Sobibor, where they both died, while his two children died at Auschwitz.[1]

Elte's semiregular polytopes of the first kindEdit

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]

Summary of the semiregular polytopes of the first kind[4]
n Elte
notation
Vertices Edges Faces Cells Facets Schläfli
symbol
Coxeter
diagram
Polyhedra (Archimedean solids)
3 tT12 18 4p3+4p6 t{3,3} CDel node 1CDel 3CDel node 1CDel 3CDel node
tC24 36 6p8+8p3 t{4,3} CDel node 1CDel 3CDel node 1CDel 4CDel node
tO24 36 6p4+8p6 t{3,4} CDel nodeCDel 3CDel node 1CDel 4CDel node 1
tD60 90 20p3+12p10 t{5,3} CDel nodeCDel 3CDel node 1CDel 5CDel node 1
tI60 90 20p6+12p5 t{3,5} CDel node 1CDel 3CDel node 1CDel 5CDel node
TT = O6 12 (4+4)p3 r{3,3} CDel nodeCDel 3CDel node 1CDel 3CDel node
CO12 24 6p4+8p3 r{3,4} CDel nodeCDel 3CDel node 1CDel 4CDel node
ID30 60 20p3+12p5 r{3,5} CDel nodeCDel 3CDel node 1CDel 5CDel node
Pq2q 4q 2pq+qp4 {}x{q} CDel node 1CDel 2CDel node 1CDel qCDel node
APq2q 4q 2pq+2qp3 sr{2,q} CDel node hCDel 2CDel node hCDel qCDel node h
semiregular 4-polytopes
4 tC510 30 (10+20)p3 5O+5T 021 CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 3CDel node
tC832 96 64p3+24p4 8CO+16T r{4,3,3} CDel nodeCDel 4CDel node 1CDel 3CDel nodeCDel 3CDel node
tC16=C24(*)48 96 96p3 (16+8)O r{3,4,3} CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 4CDel node
tC2496 288 96p3+144p4 24CO+24C r{3,4,3} CDel nodeCDel 3CDel node 1CDel 4CDel nodeCDel 3CDel node
tC600720 3600 (1200+2400)p3 600O+120I r{3,3,5} CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 5CDel node
tC1201200 3600 2400p3+720p5 120ID+600T r{5,3,3} CDel nodeCDel 5CDel node 1CDel 3CDel nodeCDel 3CDel node
HM4=C16(*)8 24 32p3 (8+8)T 111 CDel nodes 10ruCDel split2CDel nodeCDel 3CDel node
- 30 60 20p3+20p6 (5+5)tT 2t{3,3,3} CDel nodeCDel 3CDel node 1CDel 3CDel node 1CDel 3CDel node
- 288 576 192p3+144p8 (24+24)tC 2t{3,4,3} CDel nodeCDel 3CDel node 1CDel 4CDel node 1CDel 3CDel node
- 20 60 40p3+30p4 10T+20P3 t0,3{3,3,3} CDel node 1CDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node 1
- 144 576 384p3+288p4 48O+192P3 t0,3{3,4,3} CDel node 1CDel 3CDel nodeCDel 4CDel nodeCDel 3CDel node 1
- q2 2q2 q2p4+2qpq (q+q)Pq {q}x{q} CDel node 1CDel qCDel nodeCDel 2CDel node 1CDel qCDel node
semiregular 5-polytopes
5 S51 15 60 (20+60)p330T+15O 6C5+6tC5 031 CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node
S5220 90 120p330T+30O (6+6)C5 022 CDel nodeCDel 3CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 3CDel node
HM516 80 160p3(80+40)T 16C5+10C16 121 CDel nodes 10ruCDel split2CDel nodeCDel 3CDel nodeCDel 3CDel node
Cr5140 240 (80+320)p3160T+80O 32tC5+10C16 r{3,3,3,4} CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 3CDel nodeCDel 4CDel node
Cr5280 480 (320+320)p380T+200O 32tC5+10C24 2r{3,3,3,4} CDel nodeCDel 3CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 4CDel node
semiregular 6-polytopes
6 S61 (*) 041 CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node
S62 (*) 032 CDel nodeCDel 3CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node
HM632 240 640p3(160+480)T 32S5+12HM5 131 CDel nodes 10ruCDel split2CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node
V2727 216 720p31080T 72S5+27HM5 221 CDel nodea 1CDel 3aCDel nodeaCDel 3aCDel branchCDel 3aCDel nodeaCDel 3aCDel nodea
V7272 720 2160p32160T (27+27)HM6 122 CDel nodeaCDel 3aCDel nodeaCDel 3aCDel branch 01lrCDel 3aCDel nodeaCDel 3aCDel nodea
semiregular 7-polytopes
7 S71 (*) 051 CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node
S72 (*) 042 CDel nodeCDel 3CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node
S73 (*) 033 CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node
HM7(*)64 672 2240p3(560+2240)T 64S6+14HM6 141 CDel nodes 10ruCDel split2CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node
V5656 756 4032p310080T 576S6+126Cr6 321 CDel nodea 1CDel 3aCDel nodeaCDel 3aCDel nodeaCDel 3aCDel branchCDel 3aCDel nodeaCDel 3aCDel nodea
V126126 2016 10080p320160T 576S6+56V27 231 CDel nodeaCDel 3aCDel nodeaCDel 3aCDel nodeaCDel 3aCDel branchCDel 3aCDel nodeaCDel 3aCDel nodea 1
V576576 10080 40320p3(30240+20160)T 126HM6+56V72 132 CDel nodeaCDel 3aCDel nodeaCDel 3aCDel nodeaCDel 3aCDel branch 01lrCDel 3aCDel nodeaCDel 3aCDel nodea
semiregular 8-polytopes
8 S81 (*) 061 CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node
S82 (*) 052 CDel nodeCDel 3CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node
S83 (*) 043 CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node 1CDel 3CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node
HM8(*)128 1792 7168p3(1792+8960)T 128S7+16HM7 151 CDel nodes 10ruCDel split2CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel nodeCDel 3CDel node
V21602160 69120 483840p31209600T 17280S7+240V126 241 CDel nodeaCDel 3aCDel nodeaCDel 3aCDel nodeaCDel 3aCDel nodeaCDel 3aCDel branchCDel 3aCDel nodeaCDel 3aCDel nodea 1
V240240 672060480p3241920T 17280S7+2160Cr7 421 CDel nodea 1CDel 3aCDel nodeaCDel 3aCDel nodeaCDel 3aCDel nodeaCDel 3aCDel branchCDel 3aCDel nodeaCDel 3aCDel nodea
(*) 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:

NotesEdit

  1. 1.0 1.1 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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