Emanuel Lodewijk Elte

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 both died, while his two children died 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 E_n 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:

• S_n = n-simplex: S₃, S₄, S₅, S₆, S₇, S₈, ...
• M_n = n-cube= measure polytope: M₃, M₄, M₅, M₆, M₇, M₈, ...
• HM_n = n-demicube= half-measure polytope: HM₃, HM₄, M₅, M₆, HM₇, HM₈, ...
• Cr_n = n-orthoplex= cross polytope: Cr₃, Cr₄, Cr₅, Cr₆, Cr₇, Cr₈, ...

Semiregular polytopes of first order:

• V_n = semiregular polytope with n vertices

Polygons

• P_n = 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: P_n, AP_n

4-polytopes:

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

Notes