E. L. Elte

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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 kind[edit]

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 tT 12 18 4p3+4p6 t{3,3} CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
tC 24 36 6p8+8p3 t{4,3} CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
tO 24 36 6p4+8p6 t{3,4} CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
tD 60 90 20p3+12p10 t{5,3} CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node 1.png
tI 60 90 20p6+12p5 t{3,5} CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
TT = O 6 12 (4+4)p3 r{3,3} CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CO 12 24 6p4+8p3 r{3,4} CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
ID 30 60 20p3+12p5 r{3,5} CDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
Pq 2q 4q 2pq+qp4 {}x{q} CDel node 1.pngCDel 2.pngCDel node 1.pngCDel q.pngCDel node.png
APq 2q 4q 2pq+2qp3 sr{2,q} CDel node h.pngCDel 2.pngCDel node h.pngCDel q.pngCDel node h.png
semiregular 4-polytopes
4 tC5 10 30 (10+20)p3 5O+5T 021 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
tC8 32 96 64p3+24p4 8CO+16T r{4,3,3} CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
tC16=C24(*) 48 96 96p3 (16+8)O r{3,4,3} CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
tC24 96 288 96p3+144p4 24CO+24C r{3,4,3} CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
tC600 720 3600 (1200+2400)p3 600O+120I r{3,3,5} CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
tC120 1200 3600 2400p3+720p5 120ID+600T r{5,3,3} CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
HM4=C16(*) 8 24 32p3 (8+8)T 111 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
- 30 60 20p3+20p6 (5+5)tT 2t{3,3,3} CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
- 288 576 192p3+144p8 (24+24)tC 2t{3,4,3} CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
- 20 60 40p3+30p4 10T+20P3 t0,3{3,3,3} CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
- 144 576 384p3+288p4 48O+192P3 t0,3{3,4,3} CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
- q2 2q2 q2p4+2qpq (q+q)Pq {q}x{q} CDel node 1.pngCDel q.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel q.pngCDel node.png
semiregular 5-polytopes
5 S51 15 60 (20+60)p3 30T+15O 6C5+6tC5 031 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
S52 20 90 120p3 30T+30O (6+6)C5 022 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
HM5 16 80 160p3 (80+40)T 16C5+10C16 121 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Cr51 40 240 (80+320)p3 160T+80O 32tC5+10C16 r{3,3,3,4} CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Cr52 80 480 (320+320)p3 80T+200O 32tC5+10C24 2r{3,3,3,4} CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
semiregular 6-polytopes
6 S61 (*) 041 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
S62 (*) 032 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
HM6 32 240 640p3 (160+480)T 32S5+12HM5 131 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
V27 27 216 720p3 1080T 72S5+27HM5 221 CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
V72 72 720 2160p3 2160T (27+27)HM6 122 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
semiregular 7-polytopes
7 S71 (*) 051 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
S72 (*) 042 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
S73 (*) 033 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
HM7(*) 64 672 2240p3 (560+2240)T 64S6+14HM6 141 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
V56 56 756 4032p3 10080T 576S6+126Cr6 321 CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
V126 126 2016 10080p3 20160T 576S6+56V27 231 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
V576 576 10080 40320p3 (30240+20160)T 126HM6+56V72 132 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
semiregular 8-polytopes
8 S81 (*) 061 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
S82 (*) 052 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
S83 (*) 043 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
HM8(*) 128 1792 7168p3 (1792+8960)T 128S7+16HM7 151 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
V2160 2160 69120 483840p3 1209600T 17280S7+240V126 241 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
V240 240 6720 60480p3 241920T 17280S7+2160Cr7 421 CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
(*) 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:

Notes[edit]

  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