On c8rtain S8ri8s of S8ctions

of th8 R8gular Four-dirn8nsional HYP8rsolids,

BY

ALICIA BOOLE STOTT.

Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam.

(EERSTE SECTIE.)

DEEL VII. N°. 3.

(With 22 figures and 14 diagrams.)

AMS'l'ERDAM,

JOHANNES MÜLLER. 1900.

On certain Series of Sections of the Regular Four-dimensional Hypersolids,

BY

ALl CIA BOOLE STOTT.

1. In making series of seetions of the regular four-dimensional figures by the method given in this paper it is only neeessary to know the number of solids meeting at eaeh vertex. The total number in eaeh figure ean be found by eounting the · number of solids cut in the seetions.

Taking the figures bounded by tetrahedra it is evident that a seetion by a spaee eutting the edges meeting in a vertex at equal distanees from that vertex will give an equilateral triangular seetion of eaeh tetrahedroD. . Henee the complete seetion will be a three­dimensional regular figure bounded by equilateral triangles.

There are only three sueh figures, the tetrahedron, the oetahedron and the ieosahedron; so there will be no other four-dimensional figure bounded by tetrahedra ex cept those whieh have 4, 8 or 20 at eaeh vertex.

If groups of tetrahedra arranged so that. there are 4, 8 or 20 round a point he cut by parallel spaces close enough together to pass at least onee through eaeh edge, th en the number of tetra­hedra cut in the three groups respeetively will he 5, ] 6 or 600 .

N ext taking eubes. The section of a eube hy a spaee eutting the edges meeting at a vertex at equal distanees from that vertex is an equilateral triangle. So there will be no other figure bounded hy eubes exeept those having 4, 8 or 20 at eaeh vertex. But 8

4 ON CERTAIN SERIES OF SECTIONS OF THE

cubes exactly fill three-dimensional space and cannot therefore form a four-dimensiona.l angle. Hence there cannot be afigure ",hose angles are formed by eight cu bes. Still less can th ere be olle whose angles are made by 20. Similar reasoning applies to the dodecahedron; as with the cubes, there is only room round a point for four. Now the figure corresponding to the first case is the 8-cell' that corresponding to the second is the 120-cell.

Taking an octahedron. A section by a space cutting the edges . meeting at a vertex equally, is a square, and, as a cube is the only regular three-dimensional figure b.ounded by squares, there will only be one regular figure bounded by octahedra and that will have six at each vertex. This is thc24-cell.

In this manner we meet successively all the regular cells of four-dimensional space.

2. In making a series of sections of a regular four-dimensional

figure by three-dimellsional spaces 8~, 81

, etc. parallel to a bounding 1 2

solid Z, that solid itself may be considered the first element

of the series. If Z be in a space 83

, it is the only part of the figure in that space, but its faces are the surfaces of contact of it with other bounding solids. So that in building up the solids of the figure about Z in their position in four-dimensional space, if such a thing were possible, there would be one on each face. In some cases th ere are also one or more on each edge and one or more on each vertex. Whether this be so or not, can be determined by means of the number of cells meeting at each vertex in the particular figure under consideration. The solids on the faces of Z may be supposed to turn about those faces until they lie wholly

in 83

and if there be any on the edges and vertices they may be supposed to turn about those· edges ano. vertices until they too lie

in 83

. We represent in fig. 1 the result of such an operation on the 8-cell. The cube H A is the solid originally in the space

83

; NA has been turned about its surface of contact with H A, . 3

namely the square CA, into 8. The cubes P A and 8 A have been turned about the squares GAand E A respectively into the

same space d. The result of th is is that the square LA, which is common to

the two cubes N A and P A, has assumed two positions in 83

. It IS e. g. horizontal in P A and vertical in N A.

Similarly MAand 0 A each appear in two positions in 83

.

REGULAR FOUR-DlMENSION AL HYPEltSOLlDS. 5

If the 8-cell were cut by a space 83

parallel to 83

and passing 1

through some point of the edge A K, each of the cuhes NA, PA, 8 A would be cut hy a plane parallel to its surfaée of contact with H A. The positions of these planes in the cubes could easily

be determined af ter they have been brought into the space ~. For

instance, if 83

bisects A K, the sections of N A, P A and 8 A will 1 . .

be squares parallel to CA, GAand E A through the midpoint of A K. Similarly there will be square sections of the three cubes at H and the complete section is a cuhe. 'J1hus there will be three cubes in this series: 1° the cube H A, 2° a cube bounded hy the sectional planes parallel to the faces of H A and 3° a cu be bounded by the squares P K, NK, 8 K and the corresponding faces of the cubes apout II This last cube is itself a solid of thc 8-cell namely that opposite to HAl).

3. Definition l. - Let a point at a distance n times A B from A on thc line A B he the point An B or B 1 - n A.

Definition 2. -- Let pn he the projection, by a line parallel to the base on to the perpendicular of {ln equilateral triangle, of a length on the side equal to n times the side (p = t VS).

The 16-cell.

3 4. Let ABC D, a tetrahedron in 8, be one of the bounding

solids of a 16-ce11. In this figure there are 8 solids at each vertex, acondition

that will be satisfied if a tetrahedron be put on to each face of ABC D , one to each edge and one to each vertex. Let the vertices of those on ABC, B CD, C JJ A, DB A be D', A', B', C' resper.tively (fig.- 2). Then those on the edges will be AB C' D ', B CD' A', CIJA' B', IJA B' C', A CB' IJ', B IJA ' C'.

Four of the eight tetrahedra at A are represented in fig. 2.

5. If the 16-cell be cut by a space 83

parallel to 83

1

and passing through t.he point A 1 IJ', the tetrahedron ABC IJ'will "4

' ) This way of dealing with the 8-cell is given in the "Scientific Romances", No. 1, "What is th e fourth Dimension?" by C. H. Hinton,'published by Swan, Sonnenschein & Co, London, 188!, and the subsequent book of the same author: "The New Era of Thought" London, 1888. Also in considering the 600-cell I rectïved some valuable suggestions from Mr. H. W. CUljel.

6 ON CP.:RTAIN SERIES OF SECTroNS OF THE

be cut by a plane passing through that · point and parallel to ABC; hence its section wiU be an equilateral triangle with ft side equal to t A IJ'. In fig. 2 its vertices AIIJ', BlIJ', Cl IJ' are indi-

"4 "4 "4 cated as a, 0, c. Similarly the sections of B 0. IJ A', 0. IJ AB', IJ ABC' are equilateral triangles with a side equal to i A IJ' . Their vertices are the points (B IA', Cl A', IJ.1. A'), (C;! B', IJ.1 B', AlB'),

4 4 4 4 4 4 (IJL 0.', A.1. Cl, BJ 0.').

4 4 4

It has been shewn by · means of the tetrahedron AB CIJ' that g3 1

passes through the points Al IJ', 0..1. IJ ', and by the tetrahedron · 44

A CIJR' that it passes through the points Al B', Cl B'. But the edges "4 "4

AIJ', CIJ' are common to ABeD' and ACB'IJ' and the edges AB', CB' are common to ACIJB' and ACB'IJ'. Hence we have four

points on A CB' IJ' throllgh which 8: passes, giving as section a

rectangle with sides tand t of A IJ' indicated in fig. 2 by acde.

'rhere are also rectangular sections of the tetrahedra on the other edges of A BC IJ.

Again A IJ', A B', A 0.' are edges of the tetrahedl'on AB' 0.' IJ' 3

and it has been shewn that 8 cuts them in the 1

points A 1 IJ', "4

Al B', Al 0.' , gi ving as section of this tetrahedron au eq uilateral 4- "4

triangle with side equal to t A jJ' . Similarly the section (fig. 3) of each of the tetrahedra on the vertices B, 0., and IJ wiU be an equilateral triangle with sides equal to t Aj)'.

In figure 2 are also indicated the sections of the 16-cell by

spaces 83

, 83

parallel 83

and cutting A IJ' in the points A 1 IJ', 23 2

A3 D'. The shapes are shewn in the figures 4 and 5. "4 •

A space 8: parallel to 83

and passing through D' would also

pass through A' B ' and 0.', giving as the last shape of the series

a tetrahedron eq ual to A BC IJ oppositely placed, this being the bounding tetrahedron of the 16-cell opposite to ABC IJ.

The 2'-ceJl.

6. Let ABC J) 1i ]i' (fig. 6), an octahedron in a space S3, be one of the bounding solids of a 24-cell. In this figure there are 6

- solids at each vertex, a conditon that wiU be satisfied if one be

REGULAR FOUR DIMENSIONAL HYPERSOLIDS. 7

put on to each face and one to each vertex of AB CIJEF. By an inspection of figure 6 it is clear that

on the face ABC is the octahedron ABC (AB) (BC) (CA),

" " ". A CE " " " A CE (A C) (CE) (E A),

" " " AEF" " " AB F (A E) CE F) (F A),

" " " AF B " " " A FB (AF) (F B) (BA),

" " " IJBC" " " IJ B C (IJB) (B C) (CIJ),

" " " IJCE" " IJ CE (IJ C) (CE) (EIJ).,

" " IJEF" " IJ EF (IJE) (EF) (FIJ) ,

" " " IJFB" " IJ F B (IJ F) (F B) (B IJ).·

In the same manner the octahedron

on A is A (A B) (A C) (A E) (A F) A ' , " B " B (A B) (B C) (IJ B) (BF) B', " C" C (A C) (B C) (C IJ) (C E) C', " IJ " IJ (B IJ) (C IJ) (IJ E) (IJ F) IJ', " 13 " E (A E) (CE) (DE) (E F) E', " . F " F (A E) (B F) (IJ F) (E F) F ' .

. 3 3 7. lf the 24-cell be cut by a space 8

1 parallel to 8 and pas-

sing through the point of bisection of A (A C) the octahedron A (C E) wiU be cut in a plane parallel to ACE, and passing through the points of bisection of C(A C) , C(C E), E(A B), A (A E), E(CE); the section wiU he a regular hexagon (a bed e f).

Similarly the sections by 83

of the remaining seven octahedra 1

on the faces of ABC IJ E F are regular hexagons.

N ow in the octahedron A (CB) we see that 83

passes through 1

the points A 1 (A C), A 1 (A E) and in the octahedroll A (F B) we "2 "2

see that it passes through A 1 (A B) and A 1 (A F). But the lines "2 "2

A (A C), A (A E) and A (A F), A (A B) are a180 edges of the octa-

he'dron A A', whence we find that the section by S3 of this octa-1

hedron is a square with side equal to half the edge of the 24-cell. There wiU be similar square sectiûns of the octahedra on B, C, D, E, F.

The shape is shewll in fig. 7; it is a combination of octahedron and cube, in the crystallographic sense, the octahedron predominating.

Let 8: be a space paraUel to 83 and passing through (A C), it

will also pass through (C E), (A E). So that it coincides with that face of the octahedron A (C E).

8 ON CERTAIN SERIES OF SECTIONS OF THE

Tt also coincides with the face (A B) (A F) (B F) of the octa­hedron A (B F). The points (A B), (A F), (A E), (A C) are the vertices of a square section of the octahedron A A'. 'fhis . section of the 24-cell, then, is bounded by 8 equilateral triangles, · and 6 squares, the sides being equal to the edges of the · 24-cell. The shape (fig. 8) is that of the combination of octahedron and cube in equilibrium.

The octahedra already grouped about the octahedron A J) give 4 solids at (A C), namely A A', A (C E), CC', A (B C); of these the first two only are shewn in figure 6.

Putting one on the face (A C) (C E) (A E) and one on the face CA B) (A C) (B C) we have the required number 6 at (A C). There will be similarly placed octahedra at (A B), (A F), etc.

Another parallel space 83

passing through the middle point of 3

(A C) A' will give a square section of A A' and square sections of the octahedra C C' and E E' . 'fhese three squares determine the section of (A C) E'; it isa regular hexagon. This shape then

is like the section by 8:. A space . 8: parallel to . 83 and passing

through A' will coinciele with the octahedron A ' B' C' D' E' F'.

Tile 120-cell.

8. 'fhe sections of this cell can be deduced in a similar man­ner to that in which tbose of the 16-cell anel 24-cell were obtained.

'fhe plans are given in the diagrams VIII-XIV.

Tile 600-cell.

9. Let ABC D, a tetrahedron in 83

, be one of the bounding solids of a 600-cell. In this figure there are 20 solids at each vertex and this condition is satisfied at the vertices of ABC D if a tet~ahedron be put on to each of its faces, two to each edge and 10 to each vertex. The bases of the tetrahedra in each pf these groups of 20 are the faces of an icosahedron 1) . Let I A be that one which is bounded by the bases of the tetrahedra meet­ing at A. The vertex of the tetrahedron on ABC as base is .D' anel the vertex of the one on A C D' is (A C); the tetrahedron on A D' (A C) is A D' (A C) (AD)' likewise that on A (4 C) (AD) is

A (A C) (A) (A ) and that on A CA ) CA ) is A (A ) (A ) (A ). D B D IJ D B C

1) The seetions of sueh ieosahedra appear as zones on the seetions of the 600-eell.

REGULAR FOUR-DIMENSIONAL HYPERSOLIDS. 9

Thus (A) (Ac) (AD) is the face opposite to B 0 IJ on 1 A

(fig. 9 and 10), etr;: . The five tetrahedra just given are all differently related to AB 0 IJ

and, if they be taken as types, the vertices of the remaining tetra­hedra about A B 0 IJ may at once he written down and can then easily he plaeed in space. For instance there are four of the form AB 0 IJ'. 'l'he form A 0 B' (A 0) gives two tetrahedra on A 0 namely A 0 B' (A 0) and A 0 IJ' (A 0) and also two on each of the other edges as B 0 IJ' (B 0) and BOA' (B 0) on B 0, and so on. 'rhe form A IJ' (A 0) (A ) gives six tetrahedra touching A

D

arranged in paus:

A IJ' (A 0) (A ) and D

A 0' (A B) (Ac) "

A B' (A 0) (A) " B

A IJ' (A B) \A D)'

A 0' (A IJ) (A ), c

A B' (A IJ) (A ) B

and also those about B, 0, anel IJ. 80 A (A 0) (AD) (AB) gives

three tetrahedra ahout A namely A (A 0) (A ) (A ), A (A B) (A ) (A ) D n c IJ

and A (A IJ) (A ) (A ); ahout Bare B (B 0) (B ) (B ) and so on. B CAD

There are also four of the form A (A ) (A ) (A ). 'l'be remaining B c D

vertices of the 600-cell are named as follows: A is the vertex of I

the tetrahedron on (A ) (A ) (A ), (A?B) is tbe vertex of tbe tetra­B C D

hedron on (A B) (A ) (A ) and A' (A' ) (A ' ) (A' ) is related to C D n C D

A ' B (} IJ as A (AB) (Ac ) (A) is to AB O.D (see list of vertices).

The tetrahedron on the side of the 600-cell opposite to ABC IJ is a {3 7 J, the lines dra wn from A to (1" from B to {3 and so on heing diameters of the figure.

'l'he arrangement of tetrahedra ahout a {3 7 J is similar . to that about AB 0 IJ, so tbat the vertices opposite to those already given may he written down hy simply changing A into a, B into {3, 0 into 7, IJ into J.

10. Let the 600-eell he cut by a space 83

passing through the 1

points A' B' 0' IJ'. It will he parallel to S 3. Now 8~ and I A 1

interscct in a plane passing through the points B', 0', IJ', fig. 11. rl'o find wh ere this plane cuts the lines B (A B), 0 (A C), IJ (A IJ), let I A be projected on a plane passing through 0', (A ), (A 0), G.

c

10 ON CERTAIN SERIES OF SECTlONS OF THE

rl'his projection is shewn in fig. 18, wbere R, 'M, 0; Pare the points 0', (Ac )' (A 0), 0, and N is the middl~ point of (AD) (AB)'

R J7 is drawn parallel to M N and V 0 is the distance from A 0 at which the plane cuts 0 (A 0). Let V 0 be a times P 0 and we have the points (A B) B, (A C) 0, (A D) D; for the remaining ver-

a, ft {(

tices of this form see list of vertices and diagram I.

Again S~ a~d I D' intersect in a plane passing through the

points (0 A) 0, (OA) A, (A B) A, (A B) B, (B 0) B, (B 0) 0, of rr . . a a (f. a ft

fig. 12. Let R MOP, fig . 18, be a projection of 1 D' on a plane

through (A B) (D) (OD) O. rrhen R q = pa a~d q VI is parallel

to M N . It will be found that V-P = a times 0 P, whence 1

we have the points 0 (0 ), A (A ), B (B ) of fig. 12. "D t/. D (( D

From this it wiU be seen that the tetrahedron A (A ) (A ) (A ) B C 1.J

is cut in the points A (A ), A (A ), A (A ) (fig. 13, P q r); likewise ft B a C If. n

A (A 0) (A ) (A ) is cut in the points A (A ), A (A ), (A 0) A B D a B aD a

(fig. 13, P l' 8) and A D' (A 0) (A ) is cut in the points (A 0) A, D a

A (A ), D ' (fig. 13, 8 r D'). a D

'l'he tetrahedron A 0 D' (A 0) is cut in the points CA 0) A' ct

(A 0) 0, D' (fig. 13, 8 t D '). u

Here we have a section of a tetrahedron of each type given in

Fig. 10 except AB 0 D which is not cut hy s:' and as the tetra­

hedra of each type are aU similarly related to AB 0 D their sec­tions wiU be equal.

In constructing the model of this section of the 600-cell there wiU be four equilateral triangles Cp q r) related to each other as the vertices of a tetrahedron.

'rhe , sides of the equilateral triallgles are the bases , of isosceles triangles Cp l' 8, fig. 13) and the sides of these isosceles triangles are the bases of othel.' isosceles triangles (8 r D'). The last isosceles triangles are arranged in pairs with a common side, and the other sides are the sides of isosceles triangles (8 t D').

One of the four similar regions of this model is given in diagram I (N, M', L'); these regions are connected by triangles (M).

11. Let tbe 600-cell be cut by a space S3 passing through 2

(A B), (A 0), (A D), (B 0), (B D), (0 lJ). 'l'his is parallel to S3. Here 1

REGULAR :FOUR-DIMENSIONAL HYPERSOLlDS. ] 1

I (AD) and S: intersect in a plane passing through the points D'

A (A B), A (A ). a 0. C

A projection of this on a plane through A D ' (A J) Ai is given

in fig. 19. There D'?" is the projection of the plane of inter­

section of I A and S~; A r is made equal to p a and the line D' r D 1

divides (A B) A in the ratio a: I-a ; fT r is drawn through (A B) 1

and is parallel to D ' r. Then we have ?" (A ) = p a. 1 C

Also let the length of which D ' V is the projection be equal to c

ti.mes the edge, from which we see that S: passes through the points

(A ) A, (A ) A, D' (D' ),D' (D' ). In diagram II these points are C a B a c C e n ·

2, 1, 14, 11; 6 and 5 are (A B) and (A 0). A plane through I A parallel to BOD and passing through the points CA B), (A 0), (A D) gives the points D' (A ), B' (A ), 0' (A ), the first of these being

(f. D a D (t C

point 8 of diagram II, fig. 1], (see list of vertices). 'rhis model is constructed in the same way as the first. The

points where the sectional space cuts the edges of the 600-cell are detel'milled by means of the icosahedra 1) anel are then fotmd on the individual tetrahedra.

3 12. Let the GOO-cell be cut by a space s~ through(AB),(Ac),(A

D),

(B ), (B ), (B ), (0 ), (0 ), (0 ), (D ), (D ), (D ); this is parallel to S3 .4. CDA R D . l C B 2

and its intersection with I (A ) is given in projection hy the line . n V (A ) parallel to V l' (fig. 19). Now V bi se cts D'(D' ), and let

1 C "l "I C ·

(A B) Sbe equal to ftimes the edge. Again VI (Ac) divides (A B)(D'c)

in the ratio a: I-a. We have then the points (A B) (A2

B), r ?

(A 0) (I 0), (A B) (D'), (A 0) (D'), Dil (D' ), D'l (D' ); see on r IIC a 11 2 C 2 B

diagram III the points 6, 5, 13, 14, 18, 8; 1 and 2 are (A ) and . B

(A ) (consult list of vertices). C

13. Let the 600-cell be cut hy a space s:' through (A'B)' (A'e)'

(A' ), (B'), (B' ),(B' ), (0' ),(0' ),(0' ),(D' ),'(D' ),(D' );thisisparal-D A (; D .4. R D AB C

lel to S 3. 'rhe intel'section of S 3 and j (A ) is shewn in projection 4 D

by the line (D')1' parallel to V CA ), fig, 19. C 2 1 C

' ) See fcot note on page 8.

12 ON CERTAIN SERlES OF SECTIONS OF THE

Here T (A2

B) = f times the edge; r hisects (A ) A and (IJ' ) r • 2 C 1 C 2

divides(A2B) (A ) in the ratio a: I-a. C

This gives the points (A )1 A, (A )1 A, (A2B)" (A ), (A

20)" (A ),

C"2 1 B 2 1 (. B

(A2B) (A B), (ilO) (AO)~ See on diagram IV the points (5, 4, 6, 3, r r

7, 2); 8 and 1 are the points (IJ' ), (IJ' ). C B

Again I A and 83

intersect in a plane through the point 1 4

<)

(A"O) (A ) and parallel to (A ), (A ), (A ). rfhis is shewn in pro-a B D B C •

~

J'ection in fig. 22. It gives the points (A) (A (3), (A" 0) (A ), l:J a a D

(A ) (A J), (A2B) (A ).

D a a D

14. Let the 600-ce11 he cut hy a spa ce 8~ in points (A2B), (A B\

(A20), (A d) (A2

IJ), (A IJ\ (B20), (B 0\ (dIJ), (0 n\ (B·lIJ) , (B IJ\

it will he parallel to S3. The line V r , parallel t.o (IJ' ) l' and passing 2 3 C 2

~

through (A-B), (fig. 19), is the projection of the intersection of I(AD

)

and 83

. Comparing this with the line V r , we have A r = pc and 5 1 1 3

(IJ' ) 17 = P a. C 2

rfhis gives the points (A) (A ), (A) (A ), (IJ' ) (A J), (IJ' ) (A (}) J e C i e n Ca Rit

~ ?

(on plate V the points 4, 3,6, 1; 5 and 2 are (A"B) and (A"O), etc.).

The solid I A and S~ intersect in a plane passing through (A2B),

. 1 "

(A20), (A

2.D) and this plane passes through (AJ),,(A

D),(A{3))A

B),

(A'V) (A ), fig. 22. I (l C

rfhe space 83

and I Jintersect in a plane through (J)' ) (A J) that is " 1 B Cl

Parallel to (IJ' ) (IJ' ) (IJ' ). This plane is projected in the line r v, fig. 20. B A, C

Let V (IJ' A) he called h times the edge and we have the points

(IJ 'A)h(J2a), (IJ' ) (J2{3), (IJ' ) (J1y). Bh Ch

Again 83

alld I (A J) interseet in a plane through (IJ' ), (IJ' ), 4 C B

(Ab).! A1' tig. 21. Here (IJ'B )17andap~rallelplanethrough (A2B) (A

20)

is projected in the line rVi giving Al Pi = c times the edge and

(IJ' ) r = pf. Whence we have the points (A ) (A ), (IJ ' ) J, (IJ' ) J B i e IJ Br1 Cri

see list. In diagram V the zone on I A is 23, 5, 12, 2, 15, 13 and

1

that on I J1

is 2, 7, 5, 8, 9, 10, 11.

REGULAR FOUR-DlMENSIONAL HYPERSOLIDS. 13

15. Let the 600-cell be cut by a space 8~ through ABC]) 6 1 1 1 1

parallel to 83

; I(A ) intersects 83

in a plane through A parallel D 6 1

to its intersection with 83

. 5

In fig. I!} this plane is projected in the line Ai V3

and comparing " .

this with ])' r we have the points (AJ) (])' ), (AJ) (])' ), (A"B) (A 0), a· CaB a

(A? C) (A d) (diagram VI poirlts 1, 3, 4, 2, 5). Here I Jand 83

Cl < 1 6

interseet in a plane parallel to (])' ) (]) ' ) (])' ) through (AJ) (])' ). - ABC a B

This plane is projected in the line TV, fig. 20, where T (AJ) = pa. 111

Then (])') V = a times the edge, whence we have the points A 1

" ') " (D' ) (J" a), (])' ) (J-(3), (D') (J""I) (diagram VI points 12,11,13). Aa Ba Ca

Now 83

and I A interseet in a plane parallel to (A ) (A ) (A ) and 6 1 B C D

passing through the points (A2B) (AJ).

" ?

In fig. 22 the line (A"B) (AJ) is divided in the ratio a: 1-. a

and through the point of division a line r V is drawn parallel to 1

(A ) (A ) .. D B

Let (A?'B)r = 1JZ times the edge, and it will be founo. that (AJ) (A ) D

is divided in the ratio h: 1 - h, (A?()) (AJ) is divided in the ratio " " a: 1 - a, (A" C) (a' ) is divided in the ratio 1Jl : 1 --- m, (A" C) (A (3)

y

is divided in the ratio a : 1 - a ano. fillally V (A (3) = h times the 1

edge; whence we have the points

(A2

B)m (a'(3) , (A2B)a (A J), (A2B)a (A "I) , (A J)h (A) ,

(A "I)" (Ac)' (A2C)" (A J), (A

2]))a (A "I), (A'!.])t (a'd)'

(A2

C) (a'), (A2 C) (A (3), (A 2])) (A (3), (A (3) (A ).

'" y " a. h B

Here 8: and I (A d) interseet in a plane through A'I parallel to

the intersection of 83

with I (A J); Ar, fig. 21, is a projection . 5 1 2

, " of this plalle. Let (D ) r = p n. It will be found that (A" C) (a' )

B 2 Y

is cut in the ratio ?Jl: 1-111, (A? C) (i (3) is cut in the ratio C:

l-a, (D' ) (i (3) is cut in the ratio a: J-a, whence we have B

the points (A2

C) (a'), (A2B) (a' r.,)' (A

2C) (i (3), (A

2 B) (i "I),

In Y m t' C C

(])') (i (3), (])') (i'''I) , (])') J, (])') d. These are, diagram Ba Ca Bn 1 en 1

VI, the points 15, 22, 24, 23, il, 13, 26, 25, see list.

14 ON CERTAIN SERLJ<;S OF SECTIONS OF THE

16. Let the 600 ceU be cut by a space 8~ through (A (3), (Ay),

(AJ), (Ba), (By), (BJ), (Ca), (C{3), (CJ), (J)a) , (J){3) , (J)y). 'fhen I Ai

and 83

intersect in a plane thmugh (A{3), (Ay), (AJ), fig. 22, (A{3) T . 7 1

' } 'J .,

It also passes through the points (A-B) (a'), (A-C) (a') (A "D) (a',) {( (3 a y ,,0

(diagram VII points 3, 1 and 2). And 8~ and I d1

intersect in a plane

through (A J), (B J), (C J), shewn in projection by the line (A J) VZ, ? 'J '>

fig. 20. This gives the points (era) (D' ), (J-{3) (J)'), (J-y) (J)"), rt A rt n rt ç

i. e. the points 6, 5 and 4 of diagram vn. Likewise 8~ and [(A J) in­

tersect in a plane through (AZ C) (a') and ~al'allel to the in­

tersection of I (A J) anq. 8~. This flplan~ is shewn in projection by 6 ? ,) ,

the line V 1', fig 21; A V and J l' are egual top f;(A-C) (J-{3) 2 3 1 2 " 3 •

is bisected, so that we have the points (A,){ (a' ), (A) (a ' (3) , (d) (D' ), y l{ i{ lJ

(dl){ (J)'e ), (A

2C)" (a'y)' (A'~B)(t (a /(3)' (i{3)" (D'B)' (iY

2y)" (J)'e) ,

') '> 0) ') -

(I C)t (J-{3), (A-B)l (J-y). In diagram VII these points correspond to

7, 8, 12, 11, 1, 3, 5, 4, 10, 9. The symmetry shews that this is the central section; the other sections are now repeated in reverse order.

In practically constructing the sections I have found that their symmetry is made more obvious hy colouring the faces. The letters on the faces of diagrams I-XIV denote colours, tbe , plane anel accented letters denote eorresponding colours and the remaining sections are the same as those given with the plane anel accenteel letters interchanged.

17. NUll1bers of tetrahe'dra of each colour in the 600-cell as shewn by the sections.

Land L', M " M', N " N', p

" P'

R Jl' 8

" 8'

J. "

T' X Y Z

21 60 72

24

12

48 96

6

each, 42 each, ] 20

" , l4·4

" , 96

" , 48

48 .'. 1:)6

6

Total . 600

REGULAR FOUR-DIMENSIONAL HYPERSOLIDS, 15

List of the 120 vertices of the 600-cell.

A, B, C, 'D, A', B', C', IJ',

(AB), (AC), (AIJ), (BC), (BIJ), (CIJ),

(A ), (A ), (A ), (B ), (B ), (B ), (C), (C), (C), (IJ), (IJ ), (IJ ), BeD AC D AH DAB e (A'B)' (A'e)' (A'), (B'), (B'e)' (B'D)' (C'), (C'n)' (C'D)' (IJ'.,,), (IJ'n)' '(IJ'e)'

2 '2 '2 ., .} ?" "C) '2? '2 (A B), (AB), (A C), (AC-), (A-IJ), (AD-), (B-C), (BC-), (B-IJ), (BD), (C-IJ), (CIJ),

A,B,C,]), 1 i 1 1

(A{3), (Ay), (AJ), (Ba), (By), (BJ), (Ca), (C{3), (CJ), CDa), (IJ{3), (Dy),

(a'2{3), (a{3\ (a'2y ), (a/\ (a2J), (aJ'); ({32y ), (çy\ ({3'!J), ({3J\ (/J),

(a'(3)' (a), (a'd)' ({3'J, ({3'y)' ({3'd)' (y'J, (y'(3\ (y'd)' (J',,) , (J'(3)'

(a(3)' (a) (ad)" ({3J, ({3y)' ({3d), (yJ, (/(3)' (Yd)' (JJ, (J(3)'

Diagram I.

(a{3), (ay), (aJ), , ({3y), ({3J), (yJ),

a', {3', y', J', a, (3, y , J.

Ust of vertical pOints of sections. I)

A' B' C' Ij ' , (9)' (7)' (R)'

A (A ) , B (B ), C (C), D (D ), I! R (1) f'.4 ",~ ti. A

A (A ) , B (B ,),C (0 ), D (D ), (/ c (2) a (. "B ft 13

A (A ) , B (B ), C (0 ), IJ (D ), ft D (3) It D ft D n e

(AB) A , (AO) A . , (AIJ) A , (BO) B, (BD) B, (OD) 0, " (6) , ,,(,, ) ' {( (4) (( 11. ((

(AB) B ,(AO) 0 , (AD) IJ , (BO) 0, (BD) IJ, (CIJ) IJ, a .(ti) a ('12) It ('10) " a It

Diagram Il ,

(AB) , (AO) , (AIJ) , (BO), (BIJ), (OIJ), (6) (5) (I,)

?

(yJ), (J')

(Jy )' \

1) The small numbers bet ween brackets refer to the numbered points in the diagrams.

16 ON CERTAIN SERIES OF SECTIONS OF THE

(A ) A ,(B ) B, (C ) C, (D ) .D, (J a. (1) A" A a .d a

(A ). A ,(B ) B, (C ) C, (lJ ) IJ, e a (2) e IL Ban a

(A ) A ,(B ) B, (C ) C, (D ) D, iJ a (3) Jj u. iJ (I. e a

A' (B ) B' (A) C' (A) IJ' (A ) (/.. A' a B (9) " e (i;' a D '(8)'

A' (C ), B' (C), C' (B), D' (B) , [/. ,\ a B a e a D (16)

A' (IJ ), B' (D ), C' (IJ), IJ' (C) , a A a B a e ft D (17)

A' (A' ),B' (B'), C' (C', D ' (IJ') , c B c , \ c.4. c \ A ('IJ)

A' (A' ), B' (B' ) ,C' (C') ,IJ' (IJ ' ) c e c e (12) c' R (10) c B (11)'

A' (A' ) B' (B') C' 'C') IJ' (IJ' ) " D' c D (19)' c l D (18)' c e (14)'

Diagram lIl.

(AB\1 )' CB), (C), (IJ),

(Ae )(2)' (Be)' (Cn ), (IJ,),

(AD

)(3)' (B])), (C])), (IJe ),

(.4.B) (C' ), (AC) (B' ) _, (AD) (B' ), (BC) (A' ), (BIJ) (A' ), (CIJ) (A' ), a D (12) ft]) (1D) "e (10) a D a e a B

(AB) (IJ') ,(AC) (IJ') ,(AIJ) (C') ,(BC) (IJ' ), (BIJ) (C' ), (CIJ) (B' ), a e (13) a B (14) "B (11) a A a A a A

A'l(A' ), B' I(B'), C'l(C'), IJ'1.(IJ') _, 2 B 2 A 2 A 2 A (2.»

A' 1 (A' ) B' 1 (B') C' I (C') IJ' 1 (IJ' '2 e' '2 e (9)' '2' B (17)' 2' B)\8)'

A'l (A' ) B'l B' \ C'l (C') IJ'l(IJ' . '2 D' '2( D/(16)' '2 ]) (7)' 2 e\18)

Diagram IV.

(A ' ) (B') (C') (IJ' B ' A' A ' A ) (21,)'

(A' ) (B') (C' ) (IJ' e ' e (19)' B (21)' B)( 11'

(A ' ) (B') (C' (IJ ' ]) , D (1!!!' D)(22)' e )(8)'

REG ULAR FOUR·DIMENSIONAL HYPERSOLIDS. 17

(A ) 1 A (B h B (0)1 0 (lJ ) 1 IJ B '2 1(4)' A '2 l' A 2 l' A '2 l'

(A ) 1 A (B) 1 B (0) 1 0 (lJ ) 1 lJ c '2 1(5 )' C '2 ·t' R '2 l' B 2 l'

(A ) 1 A (B) 1 B (0) 1 O · (lJ ) 1 lJ D '2 1(27)' D '2 l' D '2 l' C '2 l'

(A ) (A{3) ,(B) (Ba), (0) (Ca), (lJ) (lJa) , B a (10) A a À " A a.

(Ac)" (AY)(11 )' (B c)a. (By) , (OB)} O(3), (lJ B\ (lJ{3) ,

(A n\,(AJ\9 )' (BD)a.(BJ)(25)' (OD)a.(OJ)(26)' (lJc)a.(])y)'

Diagram V.

(A2B) (AB) ,(AB

2) (AB) , r (7) r (23)

~ ~

'(IO) (A 0) (A 0-) (A 0) r (2)' r (17 )'

(A?lJ) (AlJ) ,(AlJ2

) (AlJ) , r (4) r (2~ 9 2

(BJ O),cB 0), (BO )/BO),

(B2lJ) (BlJ) , (BIl) (BlJ) , r r

9 ~

(O-lJ)r (OlJ), (OlJ-),( OlJ),

(A20) (A A

2 IJ) (A )

a B)(3)' ( a B (13)'

" ~ (A-lJ) (A (A-B) (A )

Ct C)(14)' a C (6)'

A?B A A20 (A ( )a( D)(16)' ( )a D)(12)'

(B? 0) (B), (B2lJ) (B ),

Ct A a A

" (B-lJ) (B ), Ct c

(B2A) (B ),

a D

(dB) (0 ), Ct A

(02lJ) (0 ), ft B

(dA) (0 ), a D

(lJ2B) (lJ ),

a A .)

(lJ- 0) (lJ ), a B

~

(lJJ A) a (lJ c),

2

(B A)" (Be)' ?

(B-O) (B ), a D

?

(O-lJ) (0 ), " .d

2 (OA)(O),

a B ?

(OJB) (0 ), a D

?

(lJ- 0) (IJ ), a. A

, 2 l..lJ A) (lJ ),

a B ?

(lJJB ) (lJ ). a C

(A2B) ,(A

2 0) ,(A?'lJ) ,(B?' 0), (B?'lJ), (dlJ),

(5) (2) (13)

(AB\22)' (AC?')(30)' (AlJ\17» (BO\ (BlJ\(OlJ\

] 8 ON CERTAIN SERIES OF SEOTIONS OF THE

(A1)JAB\3!' (B1)c(B ..... ), (C1)c(0.), (lJ1)c(J)),

(A1\(AC)(4 ) , (B1)c(Bc )' (C

1\(CC)' (lJ1)/lJB),

(A ) (A) ,(B) (B ), (C) (C ), (lJ) (lJ ), 1 c D (7) 1 c D 1 c D 1 c c

(A{3) (A) ,(Ba) (B ),(Ca) (C ),(lJa) (lJ), a B (23) a A a A a"'"

(Ay) (A ) _, (By) (B ), (C(3) (C ), (lJ{3) (lJ ), a C (1 ,,) a CaB a B (AJ) (A) ,(BJ) (B ), (CJ) (C ), (])y) (lJ ), a D (12) a D a D a C

2 ? ., ?

(A' ) ( (3) (B' . (3- C" -) (lJ') ( r) \ B ha, )h (a ), ( )h (ay . ' A h::t (27)'

" ? ? 2 (A' ) ( - ) (B' {3- C' (3 - lJ' ) ({3J )

C h ay, c\( y)(16)' ( )h( Y \18) ' ( Bh (11)'

(A' ) (2J) (B' ) ((32j) (C') (J 2) (lJ') ( i) D h ,Ct. '\ D h (13)' D h Y (21)' C h Y (8)'

(A' ) (Ca), (A') (lJa), B a B a

(A' ) (lJa), (A') (Ba), C a C a

(A' ) (Ba), (A') (Ca), D a D a

(B'A)a(C{3), (B')a(lJ{3),

(B'c)a(lJ{3), (B'C),.(A{3\15),

(B'D\,(A{3)(14)' (B'D),.(C{3),

(C' .J)a(By), (C')a(])Y)'

(C'B)a(])Y)' (C'B),.(AY)(19)'

(C'D)/AY\20)' (C'D)/By),

(lJ')a(BJ ) (28)' (lJ')a(CJ \26)'

(lJ' ) (CJ') (lJ') (AJ) , B a (25) , Ba ' (1) ,

(lJ C) a (AJ )(6) ' (lJ' C\ (Bd \2fl) ,

(A') (B') {3 (C' (lJ' ) J B ra

1' .4 r l' A)rY1' A r 1 (21!)'

(A'C)r a1' (B'c)r{31' (C'B)rY1' (D'B)r J1(10)'

(A'D)r a1' (B'D)r{31' (C'D)rY1' (lJ'c)r d1 (9)'

Diagram VI.

REGULAR FOUR-DIMENSIONAL HYPERSOLIDS. 19

?

(ABJa(By ),

(B20) (BJ),

a 2

(B IJ) (Ba), a

(A02) (0{3),

a '>

(BO") (OJ), a

2 (0 IJ) (Oa),

a

(AIJ2

) (IJ{3) , ct

(BD2

) (Dy), a

2 (OD) (Da),

ct

(A (3)" (A B\17)' (Ba)" (B),

(Ay),. (Ac ) (20J' (By)" (Bè), (AJ)" (An )(14J' (BJ)h (Bn ),

(A ' ) (a 2 (3),

Ba , 2

(A ) (Ct y), C a

(A' ) (a2 J), n ct .

(AB2

) (BJ), a

2 (B 0) (Ba),

a ?

(B" IJ) (By), a

?

(A 0") (0 J), a

2 (BO) (Oa),

a

(02IJ) (0{3), a

.)

(AIJ-) (1)y), a

2 (BD) (Da),

a ?

(CD) (IJ{3), a

(Oy) (0), h A

(0{3) (0), " B

(OJ) (0), " n

(Da) (IJ), " A

(IJ{3) (IJ), h B

(IJy) (D), h C

20 ON CERTAIN SERU:S OF SECTIONS OF l'HE

Diagram VII.

(Oa) (A' ), a B

(Da) (A' ), ' a n

(Da) CA' ), ct. c (Ba) (A' ), ,. c

(Ba) (A' ), " D

(Oa) (A' ), " D

(0{3) (B'), (D{3) (B' ), u A. ti A

(D{3)ct.(B'c)' (A{3)tI(B' C\:3I')'

(A{3\,(B'D)(3JJ' (O{3)"CB'D)'

(By))O'), (Dy),.(O'),

(Dy)a (C'), (Ay)ct. (0'B)(42) '

(Ay) (0') ,(By) (0' ), ct. D ('.3) ti D

(BJ) (D'), (OJ) (D'), ct. A ct. A

(OJ)a(D'B)' (AJ)(L(D'B\4)'

(AJ) (D') ,(BJ) (D' ). a C (3) a C

(A1) r (a' (3)( 8)' CA) r (a' ,."J( 7)' (A.) r (a' "\~ï)' (B

1) r ({3' ,.), (B) ,c{3' ,,), (B

1) ,c{3' ,,),

(01)/y'), (01)/'1'(3)' (OI\(y',,),

(Dt)r(J'), (D1\·(J' (3)' (D1)/JI,,) ,

(a1)r(A'B)' (at)/Al

c )" (at)/A'D)'

C(31)/B'), ({31)r(B 'c )' ({31)/B'D)'

(Yt)r(O'), (Y1)/0In)' (Yt)/O'D)'

(Jt )/D')(3j)' (d1),cD'n\12) , (Jt )/D'C\ll!'

REGULAR FOUR·DIMENSIONAL HYPERSOLIDS. 21

(6 February 1900).

A. mOlE STOlT. SecUoILS of regular four -dimensional hgpersoh'ds, I ----~-------------------------------------------------------------~

PIL

I

s

jç. .' ,

d .:~.~- :

~----, , /, -- ,

, \

\

o

\ \

\ \

\ \

b /

a.

A'

\ I , I

, I \ I

D

I I

I

I I

I

I /

I

Vérlumd.Kon.Akad, K Wetellsch.(r Sectie). Dl. VIL

C~'~ _____________ ~B'

- - --------

\ \

\ \

\ , \

\ \

\ \ \

\ \

------f---- ---­""'''-..,1

C

, , , , \

I I

I I.

CE

\\

'- ,/ V

" I: '')( I

I I

I

rïg.8. (onhaJfthe scaJ.e af FiS? 6 mul?)

9

Fig:3.

, , , , , , , ------\- -----r------

'- ' , I , I

__ ---"L- __ _

.~

e \ \

\ \ I

.,."-"",,x. ... , ...... ,/

I I

I " I .-

AD>,.' ,.,." \----

",... \ , \ \ \ \

\ \ \ \ \ \ \ \ \ \

D

FIgS.

e,

Be

Fig.13.

A(~~ ________ ~ ________ ~

A. BOOLE 51'011,' Sectiol/s ofregular four-dàneILslanal h!Jpersolids .

A

D'

\ \

\ \ \ \

flg:14 (JA.n)

, ,

, 1 1 1 1 1

:~ - - tt--

1\ , \ , ' I '-

I \ I \

I \

I

I

D~ I I I

Bo

--~ -. ---~--- .. ----- -_.---t~

Ay

/ \ I

I I , ,

--- ---y-----fAD I I I I 1

Ali

-- -------------- .--.-----t-----~- -.-

, /

/

I

\ I \ , --- :y-- - --

fAD I I I I 1

Ali

Verhand.Kon.Akad. v.J1tétensch . (1~ Sectie). Dl. FIT

, \ ,

\ , , \

, I

---1b'--IA

1''ig.18.

N

Fig.18 .

N

n' B

- -. - ~ -------'c--.----

ES: :! 1 (L IS)

PIlT

I i

i ____ _ _ J

· .. la' .L'. di n sionalllJ7'1Jerso1zds . .ABOOLE STOTT, SectJol1s ofre,su 1 lOur -· me J

IJiag. II.

-,-vJ .---------------.~ ----~--- /

8

o N I M' / 1>[ j N /

4 I

/ " 12

M

'_-~--_.--

\­

\ y

\ >; / M 1~, N .

\ M'" ,>f---6 __

"~~ 13

y

x

Pl.llI.

IhagJV:- I I

9

i5

14 y

s

A .BOOI'!; STOn; Sections ofPf.'luJar four -dim.enslonal Jl1jpel'sob.'ds . ,-------_----.::._---- ------ -------------------_.- ----

17

BiaS- V

/\ ~. I

~'2

Diag: VII.

PI IJ".

--I I

I

.. ~{ \

lvI .18 ,

~ ~ \

~N N ~.-------- \

Y ,,~ y 1';

/ \ 20

23 P P X 19

11

29 N

R

i

I 27

L_ 12

I --------------~

Verhand Kon.Akad v. Wetensch. (J~ Sectie).lJl. vJI. JBjtel./rihPJMultler; imre Leiden.

. ,f re 'u/ar fouI' L' STOTT~ ,~S:.ec=fJ_OJ_1S_Q_. _~_ A BOOLD ~

I-Diag'1f[~~\ / I

A

\ A

L

Dia.gX.

\

I

\

\ A

B

1 ersolids . -dimensional l1)P _

Dmg.XT /r--\

~ R ~ / \

B

B )

Diag.IX.

Diag-XJJ.

A

B

B

I d (je Secbe).lJ1. VIJ. d Wetens 1. Tr. Alm. v Vérl11illd. n.on.

PIV

.DjagXIII

B

~~

\ c

-~J A

B

~\ /

c

B \ / B

A

IJJ'ag.XIV

c

c