Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibór) Emanuël Lodewijk Elte at joodsmonument.nl 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.

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. 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.Coxeter, H.S.M. Regular polytopes, 3rd Edn, Dover (1973) p. 210 (11.x Historical remarks) 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 kindPage 128nEltenotationVerticesEdgesFacesCellsFacetsSchläflisymbolCoxetersymbolCoxeterdiagram Polyhedra (Archimedean solids)3 tT12 18 4p3+4p6 t{3,3} tC24 36 6p8+8p3 t{4,3} tO24 36 6p4+8p6 t{3,4} tD60 90 20p3+12p10 t{5,3} tI60 90 20p6+12p5 t{3,5} TT = O6 12 (4+4)p3 r{3,3} = {31,1} 011 CO12 24 6p4+8p3 r{3,4} ID30 60 20p3+12p5 r{3,5} Pq2q 4q 2pq+qp4 t{2,q} APq2q 4q 2pq+2qp3 s{2,2q} semiregular 4-polytopes4 tC510 30 (10+20)p3 5O+5T r{3,3,3} = {32,1} 021 tC832 96 64p3+24p4 8CO+16T r{4,3,3} tC16=C24(*)48 96 96p3 (16+8)O r{3,3,4} tC2496 288 96p3 + 144p4 24CO + 24C r{3,4,3} tC600720 3600 (1200 + 2400)p3 600O + 120I r{3,3,5} tC1201200 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-polytopes5 S51 15 60 (20+60)p330T+15O 6C5+6tC5 r{3,3,3,3} = {33,1} 031 S5220 90 120p330T+30O (6+6)C5 2r{3,3,3,3} = {32,2} 022 HM516 80 160p3(80+40)T 16C5+10C16 {3,32,1}121 Cr5140 240 (80+320)p3160T+80O 32tC5+10C16 r{3,3,3,4} Cr5280 480 (320+320)p380T+200O 32tC5+10C24 2r{3,3,3,4} semiregular 6-polytopes6 S61 (*) r{35} = {34,1} 041 S62 (*) 2r{35} = {33,2} 032 HM632 240 640p3(160+480)T 32S5+12HM5 {3,33,1} 131 V2727 216 720p31080T 72S5+27HM5 {3,3,32,1} 221 V72 72 720 2160p3 2160T (27+27)HM6 {3,32,2} 122 semiregular 7-polytopes7 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 4032p310080T 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-polytopes8 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: Sn = n-simplex: S3, S4, S5, S6, S7, S8, ... Mn = n-cube= measure polytope: M3, M4, M5, M6, M7, M8, ... HMn = n-demicube= half-measure polytope: HM3, HM4, M5, M6, HM7, HM8, ... Crn = n-orthoplex= cross polytope: Cr3, Cr4, Cr5, Cr6, Cr7, Cr8, ...

Semiregular polytopes of first order: Vn = semiregular polytope with n vertices

Polygons Pn = regular n-gon Polyhedra: Regular: T, C, O, I, D Truncated: tT, tC, tO, tI, tD Quasiregular (rectified): CO, ID Cantellated: RCO, RID Truncated quasiregular (omnitruncated): tCO, tID Prismatic: Pn, APn

4-polytopes: Cn = Regular 4-polytopes with n cells: C5, C8, C16, C24, C120, C600 Rectified: tC5, tC8, tC16, tC24, tC120, tC600

See also Gosset–Elte figures

Notes