Integral Cayley numbers

INTEGRAL CAYLEY NUMBERS

BY H. S. M. COXETER

The first part of this paper contains a simple proof of the famous "eightsquare theorem" which expresses the product of any two sums of eight squaresas a single sum of eight squares (2). This proof, like that of Dickson [8; 158-159], employs the system of Cayley numbers, which is somewhat analogous tothe field of complex numbers and to the quasi-field of quaternions.

Since the integral complex numbers and integral quaternions were exten-sively studied by Gauss and Hurwitz (see 3), it is strange that the analogousset of Cayley numbers has been comparatively neglected. There is a shortpaper by Kirmse [14], but that is marred by a rather serious error, as we shallsee in 4. Bruck showed me a simple way to remedy this defect. 5-12 areconcerned with a verification that Kirmse’s set of Cayley numbers, as correctedby Bruck, is indeed a set of integral elements, according to a precise definitiondue to Dickson. A geometrical representation is found to be helpful (6), andit appears that the integral Cayley numbers correspond to the closest packingof spheres in eight dimensions, just as the integral quaternions correspond tothe closest packing in four dimensions.

I wish to thank H. G. Forder (of University College, Auckland, N. Z.) forarousing my interest in Cayley numbers, and R. H. Bruck (of the Universityof Wisconsin) for pointing the way out of the impasse into which Kirmse’smistake had led me.

1. Cayley numbers" associative and anti-associative triads. In the notationof Cartan and Schouten [1; 944] the Cayley numbers are sets of eight real num-bers

[ao,al,a.,a3, a4,a,as,

which are added like vectors and multiplied according to the rules

er --1, er+ler+3 er+2er+6 er+4er+5 er

er+3er+l er+6er+2 er+ser+4 --er er+7 er

These rules may be written in the concise form

(1.1) e eee eee eaee eee eee eee eee --1,

provided we interpret ".-e -1" to mean

e2e4 --e4e2 el e4e --ele4 e2 vie2 --e2el e4

like the famous relations i j2 ]c2 ij -1 of Hamilton [12; 339].

Received July 6, 1946.

561

562 H. S. M. COXETER

The seven associative triads

correspond to the lines of the finite projective geometry of seven lines and sevenpoints, represented diagrammatically in Fig. 1. Thus any operation of thesimple 168-group generated by the permutations

(1 2)(3 6) and (1 2 3 4 5 6 7)

will leave the multiplication table unchanged, provided we make suitable changesof sign. (The relevance of this group seems to have been first noticed by Mathieu[16; 354].) Moreover, we may change the sign of all the e’s except those of anassociative triad, without making any permutation. Such changes of sign form,with the identity, an Abelian group of order 8 and type (1, 1, 1). Hence thesymmetry group of the multiplication table is of order 8.168 1344, and theabove is one of

27 7!/1344 480

possible notations; so it is not surprising that every author uses a differentnotation.

Fig. 1

Although the associative law holds for the triads in (1.1) and for any triadwith a repeated element, such as

ele2 --e2 e .ele2 or ele2"e e2 el .e2el

it does not hold universally. In fact, the remaining 28 triads are anti-associative"

ee2.ea --e --e.e2e3 ee2.e5 e7 --e.ee,

INTEGRAL CAYLEY NUMBERS 563

and so on, using the cyclic permutation (1 2 3 4 5 6 7).the two kinds of triad was pointed out by Cayley [3].)

(The distinction between

2. The alternating-associative law and the Eight Square Theorem.Cayley number a a0 glel - aTeT, we define the conjugate

For any

a ao ale1 aTe7

the norm

Na a a a + a -- -- a,and, if a 0, the inverse

-1 (Na)-l,a

which has the property a-la aa-1 1. It is easily verified (as in the caseof quaternions) that a- d, but it is not so easy to deduce that

(2.1) Na.Nb N(ab).

As a first step, we prove that the associative law holds for dab. (See Kirmse[14; 67].) When written out in full, the expressions da.b and d.ab agree termby term, except possibly for terms such as

ae ae. brer

where %e,er is an anti-associative triad. But they actually agree here too; for,

e,,eq.er --eq%’er eq’%er

and therefore ae,ae, be, a,e. a,%be (The same argument showsthatab a.ab.) Similarly

ab.b a.bb.

From the identity da.b d.ab, multiplication by (Na) -1

b a-l.ab.

gives

Similarly ab.b-1 a. Hence

(ab)- b-. b(ab)- b-. (a-i.ab)(ab)-1 b-a-.

(See Moufng [18; 418].)We are now ready to prove (2.1). If a or b vnishes, it is obvious.

we have

N(ab) ab ab b-d.ab (Nb. b-i.Na.a-)ab

If not,

Nb.Na.b-a’-.ab Na.Nb.(ab)-ab Na.Nb.

564 H.S.M. COXETER

When written out in full, (2.1) is the Eight Square Theorem"

(a -I-al - __aT)(bo + bl + -- b)

(aobo ab a.b2 ab ab ab ab a767)

+ (aob % abo % a2b + azb ab2 W ab ab aTb),where the implies summation of seven squares given by cyclic permutationof the suffix numbers 1, 2, 3, 4, 5, 6, 7, leving 0 unchanged. This was firstgiven (in a less systematic notation) by Degen in 1818 [8; 164].

3. The arithmetic of an algebra. The complex number a ao ali, thequaternion a ao - ali a.j a3k, and the Cayley number

a ao + ale1 + a2e2 W a3ea + a4e4 + ase + a6e6 + aTeT,

all satisfy a "rank equation" with real coefficients:

x 2aox% Na 0

(2ao a + , Na a).

A set of elements selected from an algebra is called a set of integral elements[10; 141-142] if it satisfies the following four conditions"

(i) For each element, the coefficients of the rank equation are integers,(ii) The set is closed under subtraction and multiplication,(iii) The set contains 1,(iv) The set is not a subset of a larger set satisfying (i), (ii), (iii).According to this definition, the integral elements of the algebra of complex

numbers are the Gaussian integers ao - ali, where ao and al are ordinary integers.These are represented in the Argand diagram by. the ordinary lattice points,which are the vertices of the regular tessellation of squares, {4, 4}. The units,i.e., elements of unit norm, are 1 and i, and the corresponding points

(+/-, 0), (0, =)

form a square: the vertex figure of the tessellation, as indicated by the second 4in the Schlifli symbol {4, 4}.

Analogously, the integral quaternions are ao ali aj atc, where thefour a’s are either integers or all halves of odd integers. These have been in-vestigated by Hurwitz [13]. The 24 units

1, i, -+-j, -+-k, 1/2(+/- 1-+-ij-+-/c)

are the elements of the "binary tetrahedral group" [6; 372]. If we representthe quaternion ao -+- ai a2j - atc by the point (ao, al, a, a.) in Euclidean4-space, we find that the 24 units are represented by the vertices

(:, o, o,o), (0, :i: 1, O, 0), (0, 0, =i=1, 0), (0, O, O, :t: 1),

INTEGRAL CYLEY NUMBERS 565

of the 24-cell {3, 4, 3}, one of the regular polytopes discovered by Schlfli[19; 51] about 1850. Hence the integral quaternions are represented by thevertices of the regular honeycomb {3, 3, 4, 3}, whose vertex figure is {3, 4, 3 }.A typical cell of the polytope {3, 4, 3} is the octahedron {3, 4} whose vertices

represent the six quaternions 1, i, 1/2(1 W i =i= j =t=/c). A tpical cell of the honey-comb {3, 3, 4, 3} is the cross polytope {3, 3, 4} whose eight vertices representthese same quaternions along with 0 and 1 i.

In terms of the combinations

1/2(1 -k), l. 1/2(1- /), 13 1/2(i-t-j), 14 1/2(i- j),

of norm 1/2, we have

1 11 + 12, i 13W 14, j 13- 14, k 11- 12,

1/2(1 WiWj+]c) 11+ 13.

By addition and subtraction we can derive all expressions of the form :t:l, :t:l..Hence these (with r s) are the 24 units, and the integral quaternions consist ofall expressions

where the x’s are integers and x is even. In other words, the vertices of{3, 3, 4, 3} (of edge 2t) admit as coordinates all sets of four integers having aneven sum [4; 347]. The eight Cayley numbers represented by a typical cellmay now be expressed in the simple form

l :i: 1. (s 1, 2, 3, 4).

Hurwitz’s integral quaternions evidently satisfy Dickson’s conditions (i)-(iii).Since they are closed under subtraction, we see that the vertices of {3, 3, 4, 3}form a lattice. Condition (iv) may be verified geometrically as follows. A pointsituated as far as possible from the nearest vertex of {3, 3, 4, 3} must be at thecenter of a cell. Now, the circum-radius of {3, 3, 4} (of unit edge) is 2-1/2; e.g.,this is the distance between the points representing 0 and ll Since 2-t 1,we cannot add further points to the lattice without reducing the minimum dis-tance between pairs of points.The same argument in two dimensions (where the cell is a square of circum-

radius 2-1/2) explains why the integral complex numbers ao - ali admit only in-tegral values for ao and a. On the other hand, the quaternions ao - aia2j - ak with integral a’s are represented by the four-dimensi6nal cubic lattice14, 3, 3, 4}; and this can be augmented, since the circum-radius of the hyper-cube4, 3, 3 is exactly 1.

4. Kirmses mistake. Integral Cayley numbers have been discussed byKirmse, whose notation is related to that of Cartan and Schouten (1.1) as follows:

566 H.S.M. COXETER

Thus Kirmse [14; 64] considers Cayley numbers ao -- alix -- -- a7i7, where

(4.1) i2r iii ilii iaii iii7 i2ii iiii i2ii7 --1.

From these he selects an eight-dimensional module, i.e., a set of rational Cayleynumbers which is closed under subtraction and contains eight linearly inde-pendent members. As usual, a module is called an integral domain if it is closedunder multiplication. A simple instance is the module J0 consisting of allCayley numbers

’0 - "1i1 2[_ _[_ "y7i7

where the ,’s are integers [14; 68]. Hethen defines a maximal (umfassendste)integral domain over Jo as an extension of Jo which cannot be further extendedwithout ceasing to be an integral domain. He states that there are eight suchdomains, one of which he calls J and describes in detail [14; 70]. Actuallythere are only seven, which presumably are the remaining seven of his eight.We easily verify that J itself is not closed under multiplication. In fact, theproduct

1/2(1 + ix - i -b i3)" 21-(1 + i + i - i) 1/2(i1 -{- i2 -{- i -is not one of the 240 "units" [14; 76]. When I pointed this out to Bruck, hesent me a revised description of such a domain.

Bruck’s domain J can be derived from Ji by transposing two of the i’s. Weshall verify in 5 that it is closed under multiplication, and in 11 that it is maxi-mal. Since the 168-group (l) is doubly transitive on the seven i’s, any trans-position will serve to rectify J in the desired manner. But there are only

seven such domains, since the (72)= 21 possible transpositions fall into 7sets

of 3, each set having the same effect. In each of the seven domains, one of thei’s plays a special role, viz., that one which is not affected by any of the threetranspositions.Comparing Kirmse’s multiplication table (4.1) with Cayley’s

(4.2) i ixi2i3 ixii i3ii7 i3Qi i2ii7 ixiTi6 i2i4i6 --1

[2], we see that either can be derived from the other by interchanging i withi7 nd reversing the sign of i Accordingly, Kirmse’s J could be used as itstands if we replaced his multiplication table by Cayley’s. In that case thespecial unit would be i but we shall find it more convenient to specialize i.

5. The integral domain J and its 240 units. The relation between the nota-tions of Dickson, Cayley, and Cartan-Schouten may be expressed as follows:

i= i =e2, j i2 =e, k i3 -e4, e i4 =e3,

ie i5 es je i6 eT ke i e6


Thus ao - ai -- + ai q - Qe, where q and Q are quaternions:

In other words [8; 158], Cayley numbers are derived from quaternions by ad-ioining a new unit e, which enters into multiplication according to the rule

(q - Qe)(r + Re) qr RQ + (Rq - Q)e,

where the quatrnions and are the conjugates of R and r.Now, the three Cayley numbers

i, j, and h=1/2(i-j-k-e)

generate (by multiplication) k ij and ih, jh, kh.the module J based on the eight Cayley numbers

We proceed to verify that

(5.1) 1, i, j, t, h, ih, jh, kh

is closed under multiplication. In fact,. k h=3 1,

jk -tj= -ih.h i, ki -ik -jh.h =j, ij= -ji= -h.h l,

i. ih j.jh tc kh h,

ih.i -h.ih h- i, jh.j -h.jh h- j, kh.tc -h.lh-- h- k,

(ih) hi -1 ih, (jh) hj -1 jh, (Ich) hk -1 kh,

jh.lc -j ih, kh.i -to jh, ih.j -i kh,

kh.j -tc + ih, ih.t -i - jh, jh.i -j + kh,

jh.ih t- h-ih, h.jh i- h-jh, ih.kh =j- h- tch,

kh.ih -j + h- ih, ih.jh -k + h -jh, jh.kh -i + h- kh,

tc.jh -j.kh 1 + j- t + ih, i.tch -tc.ih 1+]c- i W jh,

j.ih -i.jh 1 +i-j+

The accuracy of this table may be efficiently checked by applying the asso-ciative law to three elements of which two are equal; e.g., (k.jh)k k(jh.k).It is interesting to observe also that

(ih)3= (jh) (lh)3 1,

and that the conjugates of h and ih are -h and -ih 1.

568 H.S.M. COXETER

This integral domain J includes the elements

e 2h-i-j- k, ie 2ih+ l+j-

je 2jh W1W k i,

1/2(-1 -j + k + ie) ih,

1/2(-1 k W i + je) jh,

1/2(-1 i + j W ke) kh,

1/2(-l+ie+je+ke)

ke 2kh + 1+i-j,

1/2(i W e W je Ice) h + jh kh i,

1/2(j + e + ke ie) h W kh ih j,

1/2(k + e W ie je) h W ih jh k,

all of which are of unit norm.ke, we find altogether 240 units:

.4.1, +/-i, +/-j,

1/2(.4.1 .4- j .4- k .4- ie),

1/2(.4.1 -4- k .4- i .4- je),

1/2(.4.1 .4- i .4- j .4- ke)’,1/2(-4-1 .4- ie .4- je .4- ke),

1/2(.4.1 -4- i -4- e -4- ie)

1/2(.4-1 .4- j .4- e .4- je),

1/2(.4.1 .4- k .4- e .4- ke)

(See Kirmse [14; 76].)

By adding or subtracting 1, i, j, k, e, ie, je or

1/2(+/-i .4- e .4- # -+- ke),

1/2(+/-j .4- e .4- ke -4- ie),

1/2(+/-k +/- e +/- ie #),

1/2(+/- i .4- j -4- k .4- e)

1/2(+/-j +/- =e je +/- e),

1/2(+/-k .4- i -4- ke .4- ie),

1/2(+/-i .4- j .4- ie -+-je).

The last seven rows of this table have the followingproperties: two elements in the same row have no common terms, but any twoelements not in the same row have just two common terms, and the four re-maining terms of two such elements form another element of the set. Since allthe basic units (5.1) are included, it follows that only three types of Cayleynumber

a ao +ali + aj + ak + a,e + aie + aje + aTkecan occur in J: the constituents ao al aT may be integers, or four oreight of them may be halves of odd integers. Moreover, if four of them arehalves of odd integers, these must be distributed as in the above table. Inparticular, this shows that the table is complete: J contains only 240 units.

1+ ih + jh + kh, 1/2(i W j + k + e) h,

1/2(-l + i + e + ie) h + ih k, 1/2(-j k + je ke) jh kh i,

1/2(-1 + j + e + je) h + jh i, 1/2(-k i + ke ie) kh ih j,

1/2(--1 + k T e T ke)= h + kh j, 1/2(-i j + ie je) ih jh k,


6. A geometrical representation. For further investigation of the integraldomain J, we shall find it convenient to replace the basis (5.1) by

(6.1) 1, j, e, ke, -h, ih, jh, eh,

where, as before, h 1/2(i -- j - / -t- e). These eight units will serve equallywell, since

kh -ih jh eh 2,

i =j-le- 2kh- 1 nd

k=2h-i-j-e.

Their special virtue lies in the possibility of associating them with the nodesof a certain tree (Fig. 2) in such a way that two of them, say a and b, satisfy

N(a + b) 1 or 2

according as the corresponding nodes are adjacent or non-adjacent.

e -h j ih 1 eh ke

ljh

Fig. 2

Let us see what this means geometrically, when we represent each Cayleynumber by the point in Euclidean 8-space whose Cartesian coordinates are itseight constituents (i.e., the coefficients of 1, i, j, k, e, ie, je, ke). The domainJ is now represented by a certain lattice, which we shall identify with thevertices of a uniform honeycomb called 52.

Since the origin represents the Cayley number 0, we can equally well saythat we are representing the Cayley number a by a vector a of magnitude(Na). Then the unit vectors representing (5.1) or (6.1) generate the lattice.Two unit vectors a and b make an angle whose cosine is the scalar part of gb, i.e.,

cos (ab) aobo + abl - aTb7 1/2(-db ba).

Since N(a - b) (a + ) (a -- b) Na -b ab + a -- Nb ab + a -- 2,we have (ab) ]r or 1/2r according as N(a -- b) 1 or 2. In the latter caseit is natural to speak of a and b as orthogonal Cayley numbers.More generally, two Cayley numbers a and b (of rbitrary norms) are said

to be orthogonal whenever ab -t- a 0. Then

N(xa - x2b) N(xla) + N(xb)

for any ordinary numbers x and x2.

570 H.S.M. COXETER

For use later on, let us see whether we can find a unit of J orthogonal toof (6.1) except e. (This would enable us to add an extra branch and node toFig. 2 on the left.) Being orthogonal to 1, j and ke, it must have the form

1/2(q- k e +/- ie - je).

Being orthogonal to h, ih, jh, eh, it must have opposite signs for l and e, oppositesigns for/ and ie, the same sign for k and je, opposite signs for ie and je. Henceit is either

(6.2) 1/2( e-- ie -b je) 21c h- ih --jh

or the negative of this.

7. Reflections in eight-dimensional Euclidean space. Let A or Aa denotethe operation of multiplying every Cayley number on the left or right (re-spectively) by a Cayley number a of unit norm. These are congruent trans-formations of the Euclidean 8-space, since the squared distance between thepoints representing two Cayley numbers x and y is N(y x), which is the sameas N(ay ax) or N(ya xa) in virtue of (2.1). (The corresponding trans-formations of elliptic 7-space were investigated by Cartan and Schouten [1].)

Let o denote the reflection in the hyperplane perpendicular to the unitvector a. Since x and - differ only in the sign of their scalar parts, )1 changesevery x into -. Since a 1 a (meaning the vector 1 operated on by /a),the principle of transformation gives

which changes x into -xd a -(a})a. Similarly a 1 ’, and A,A,which changes x into -a dx -a(a). Thus the associative law holds for anyproduct of the form aba (with Na 1, and consequently also without thisrestriction; see [17; 216]), nd we see that is the transformation

(7.1) x -aSa.

The following lternative proof is suggested by Witt’s treatment of quater-nions [20; 308]. The reflection transforms any vector x into x 2(x.a)a.It therefore transforms the corresponding Cayley number x into

x (x + a)a --(a)a (Na 1).

Since (x.a)a a(a.x), we may equally well write this as

x a(x -- a) --a(a).

8. The polytope 421 The domain J is evidently invariant under the trans-formation (7.1) when a is any one of the 240 units; in particular, when a is oneof the basic units (6.1). We are thus led to consider reflections in the hyperplanesthrough the origin perpendicular to the corresponding eight vectors. Of the


28 dihedral angles between pairs of these eight hyperplanes, seven arearranged as in Fig. 2, while the remaining 21 are right angles. The eight re-flections generate a group known as [3"2’1]. (The indices are the numbers ofbranches of the tree that emanate from the central node 1 in various directions.)

Fig. 3

Fig. 3 (or briefly, 421) is the symbol for the eight-dimensional polytope whichhas one vertex on the line of intersection of seven of the eight hyperplanes(viz., all except the one whose node is ringed) while its remaining vertices arethe transforms of this one by the group [3’2’1]. The position of the first vertexis determined by the Cayley number (6.2), orthogonal to ll of (6.1) except e.Since this is one of the 240 units, we conclude that all the vertices of 421 representunits of J. But 421 is known to have 240 vertices [7; 11.8]. Hence the 240vertices of 421 represent the 240 units of J.

9. The honeycomb 521 We can go further, and identify the lattice of Jwith the vertices of the eight-dimensional honeycomb 521 whose vertex figureis 4.1 The eight hyperplanes indicated by the nodes in Fig. 2 or 3 form anangular region at the origin (like a trihedron in ordinary space). This is afundamental region for the finite group [3’2’1]. The fundamental region for theanalogous infinite group [35’2’1] is the simplex cut off from this angular regionby a hyperplane perpendicular to the direction of (6.2) but not through theorigin. The corresponding node of the tree is ringed in Fig. 4. The point ofintersection of the first eight hyperplanes is, of course, the origin. This is onevertex of the honeycomb 521 and the remaining vertices are derived from itby applying [35’2’1]. One of them is the point representing 2k h ih A- jh,provided we take the ninth hyperplane to lie midway between this and theorigin. This point is one of 240, derived from it by means of the subgroup[3’2’1]. These are the vertices of 41 (representing the 240 units) and terminatethe 240 edges of 521 that radiate from the origin. Each of these edges is per-pendicularly bisected by a hyperplane of symmetry, the transform of the "ninthhyperplane" by some element of [3’2’1]. There is also, perpendicular to eachedge, a hyperplane of symmetry through the origin. The product of the re-flections in these two parallel hyperplanes is just the translation along the edge.Thus all such translations are elements of the group [3a’’1]. The eight edgesterminated by the points representing (6.1) yield eight translations whichgenerate the lattice of J. Hence every element of J is represented by a vertexof 5.1 Conversely, every vertex of 52 represents an element of J; for, thelattice includes all the 240 neighbors of the origin, and so also all the 240 neighborsof any other vertex.

572 H.S.M. COXETER

The consequent coordinates for the vertices of 52 were first obtained byDu Val [5; 185].

2k h ih - jh e -h j ih 1 eh ke(R) : ; "-’" i’

:

1jh

Fig. 4

The polytope 4. was discover,ed by Gosset in 1897. it has cells of two kinds,whose symbols are derived from Fig. 3 by omitting one of the terminal nodes(with its branch)" a seven-dimensional regular simplex a7, by omitting the nodemarked jh in Fig. 2, and a seven-dimensional "cross polytope" (or octahedron-analogue) /7 by omitting the node marked ke. Actually [11; 48] there are17280 a’s and 2160 7’s, so that the order of the symmetry group [34’2’1] is

17280.8! 192.10!.

Analogously, the honeycomb 5z has cells as and s, whose symbols are de-rived by omitting (in turn) these same nodes from Fig. 4. Each vertex is sur-rounded by 17280 as’s and 2160 fs’s.

10. An alternative notation.

11 I(1 A-e),

12 1/2(l--e),

of norm 1/2, we have

1 lA- l.,

In terms of the combinations

1/2( + e),. 1/2(i e),

i= 13A-14,

h 1/2(11 12 A- 13 -4- 14 A- l A- 16 A- l -t- /s),

ih 1/2(-l l A- la 14 l 16 A- l A- /s),

jh 1/2(-l l + 1-4- 14-4- l- l- l- /s),

kh 1/2(-l l. 13 14 A- la A- l -4- l /s).

By addition and subtraction we can derive all expressions of the form =t= lr =t= 1,,and also 1/2(=t=11 =t= l =t= l :t= 14 =t= l =t= l =t= l =t= /s) with any odd number ofminus signs. Hence these (with r s) are the 112 -f- 128 units, and J consists

of all expressions

(10.1) xll -4- -4- xsls

where the x’s are either eight integers with an even sum or eight halves of odd integerswith an odd sum. In other words, the vertices of 5. (of edge 2t) admit as co-

j l -f- l, k l A- ls,


ordinates all sets of eight integers with an even sum, or eight halves of odd in-tegers with an odd sum [4; 385]. These are rectangular Cartesian coordinates;for, since the l’s are mutually orthogonal,

N(xlll + - xsls) N(xl/1) + + N(xs/s) 1/2(x - + x).

A typical cell as represents the nine Cayley numbers

1/2(,- . ), ,- . (s , 2, ..., 8);

and a typical cell Bs represents the sixteen Cayley numbers

ll.(Note that this as and 8 have a common a7 .)

Defining E l. 1/2(1 - e), so that

l E, 13 iE, 15 jE, 17l. E, 14 iE, 16 jE, ls

we may express (10.1) as

(x, + xi - xj - xk)E + (x2 - x4i + x6j + xsk)E.

Every product of two l’s is half one of the units =i=l =t= 1. in fact, we have

EE =EE=-(E) -(E)-E -E iE’iE -iE’iE 1/2e,

E iE iE E iE E E iE jE kE kE .jE jE kE

-kE.jE 1/2i,

iE.E E.iE -E.iE -iE.E -jS. kE tE.jE =. jE. kE

-E.iE 1/2ie,

and other rules derived from these by cyclic permutation of i, j, k.

11. The interstices of the lattice. A point situated as far as possible fromthe nearest vertex of 5. must be at the center of a cell. Since the circum-radiiof as and 8 (of unit edge) are - and 2-1/2, the latter being the greater, such a pointis actually at the center of a 8, and its distance from the nearest lattice point(or rather, from the sixteen nearest lattice points) is 2-t. In fact, this is thedistance from the center l of the cell l =t= l. to any of its vertices, such as 0.Hence every point in the 8-space is distant 2-1/2 or less from some vertex of 5

12. Dicksoa’s citerion. We are now ready to verify that J is a set of integralelements according to Dickson’s definition (see 3). We consider the four con-ditions in turn.

574 H.S.M. COXETER

(i) For each type of element (5.2) we observe that 2ao and Na are integers.(ii) Being a module, J is closed under subtraction; and we saw in 5 that it

is closed under multiplication as well.(iii) The element 1 occurs in the basis (5.1).(iv) By 11, since 2-t < 1, we cannot add further points to the lattice of J

without reducing the minimum distance between pairs of points.

13. Integral Cayley numbers of norm 1, 2, 3 or 4. As we saw in 8, the 240integral Cayley numbers of norm 1 are represented by the 240 vertices of(Fig. 3), or let us simply say that these units are the 240 vertices of 41 Inthe notation of 10, they consist of

112 like 4-11 4- l and 128 like 1/2(-/1 l - 13 - -t- l)

(with an odd number of minus signs). Similarly, the integral Cayley numbersof norm 2 are

16 like 4.2ll 1120 like 4.1 4. l -+- 13 4. 14,

and 1024 like 1/2(3/ + l + l(with an even number of minus signs); those of norm 3 are

1344 like 4.2/1 4- l 4. 1., 1792 like 4-11 4. l 4- l 4. 14 4. 15 4.

and 3584 like 1/2(3/1 -t- 312 13 14

(with an odd number of minus signs); and those of norm 4 are the doubles ofthe 240 units and also

8960 like 4-2/1 -+- l 4. 1. 4. 14 4- 15,

128 like 11 12 - ls

(with an even number of minus signs),

7168 like 1/2(3/1 + 31 + 31 + l - -t- ls)

(with an even number of minus signs), and

1024 like 1/2(-5/, -t- l - l + -t-(with an odd number of minus signs).

Fig. 6

Fig. 5

Fig. 7


A typical cell a7 of 41 has the 8 vertices

and a typical cell 7 has the 14 vertices

11 ls (s 2, 8);

11-+- ls (s 2, ..., 8).

(All the other cells could be derived from these by applying the group [34’2’1].)Since the center of this f is 11 the 2160 integral Cayley numbers of norm 2[14; 76] are the centers of the 2160 7’s of a 4.1 of edge 2, or the 2160 verticesof a 241 (Fig. 5). Moreover, since the mean of the two Cayley numbers 11 + land 11 + 13 is 1/2(2ll + 12 + 13), the 6720 integral Cayley numbers of norm 3 arethe mid-points of the 6720 edges of that same 421 or the 6720 vertices of the"truncated 421 whose symbol is shown in Fig. 6. Again, since the center of theabove a7 is

the 17280 integral Cayley numbers of norm 4 (other than the doubles of thoseof norm 1) are the centers of the 17280 aT’s of a 4.1 of edge 8/3, or the 17280vertices of 14 (Fig. 7). (If we went further in this direction we would obtainsymbols having more than one ringed node.)A still more convenient notation, for some purposes, is obtained by defining

x, (s 1, 8)yo---- xr, y--x

and lo 1/2(1 +i+j+ k) 1/2 ’lr,

so that

Z:u 0, x. u. + 1/2Uo,

and

Since y. x. 1/2 x + 1/2 x, where x. 1/2 x is an integer for everyCayley number (10.1), the nine y’s are either integers, or integers +1/2, or in-tegers --. Conversely, such y’s (with sum zero) lead to values of x. y. -t- 1/2yowhich are either integers with an even sum 3yo or halves of odd integers with anodd sum 3yo Hence J consists of all expressions

yolo -]" ylll + + y818

576 H.S.M. COXETER

where the nine y’s have sum zero and are either integers .or integers plus or integersminus .

Moreover, since

we have

+ 1/2 o) + E +

N(yo/o + ylll + + ysls) 1/2(yo + y + -+- y),

and the above sets of y’s are Cartesian coordinates for the vertices of 5.1 (ofedge 2t) in the 8-space yo yl + W y8 0 (See [4; 393]).Thus the 240 integral Cayley numbers of norm 1 consist of

( 8)72 likelo- 11 and 168 like+/- l0 + l + l.--- lr

Since lr 3/o, these may be expressed as

lr l, and +/-(l + l, + It lo),

where r, s, take any three distinct values among 0, 1, 8.the 2160 integral Cayley numbers of norm 2 are

Similarly [4; 397],

1,, + l lr l., +/-(2/,. + l. lo)and +/-(l. + l + + l. l, lo).

(756-+- 144-t- 1260 2160.)Perhaps the greatest advantage of this notation is that it enables us to replace

the nine Cayley numbers named in Fig. 4 by

14. Postscript. After I had written the above, Olga Taussky Todd drew myattention to a recent paper by Mahler [15], which refers to Dickson [9]. I thusbecame aware that Dickson himself had discovered a system of integral Cayleynumbers as long ago as 1923. As we saw in 4, the seven possible systems maybe described as specializing (in turn) the seven units i, j, k, e, ie, je, ke, ori, i, i3, Q, Q, Q, i 7. In defining J, I chose to specialize e, thus preserving thecyclic symmetry of i, j, k. Dickson obscured this symmetry by specializing i(or j or k) instead of e. In fact, he asserted [9; 324] that there are only threesystems which include all the eight basic units and satisfy the conditions for aset of integral elements. He missed the remaining four of the seven systemsby adding the very artificial requirement [9; 321] that the integral Cayley num-bers should include the integral quaternions 1/2(+/-1 +/- i +/- j +/- k).

Possibly Mahler’s work would have been simplified by specializing e insteadof i. He says the following theorem [15; 127] is "not easy to prove:"


For every Cayley number X there is an integral Cayley number G such thatN(X G) _< 1/2.

Dickson had succeeded only in proving the statement with 5/4 in place of 1/2.We now recognize this refinement as an immediate consequence of 11.

Finally, Mahler proved that every left (or right) ideal in the ring of all integralCayley numbers is a principal ideal, generated by a multiple (in the ordinarysense) of an integral Cayley number of norm 1 or 2 or 4; but not every integralCayley number of norm 2 or 4 will serve. It would be interesting to find outwhat subsets of the 2160 of norm 2 and of the 17280 of norm 4 are involved here,and what polytopes are formed by the corresponding sets of points in eight-dimensional space.

REFERENCES

1. ELm ChTAN AND J. A. SCHOUTEN, On Riemannian geometries admitting an absoluteparallelism, Koninklijke Akdemie van Wetenschppen te Amsterdam, Proceedingsof the Section of Sciences, vol. 29(1926), pp. 933-946.

2. AaTHUR CAVLEV, On Jacobi’s elliptic functions, in reply to the Rev. B. Bronwin; and onquaternions, The Collected Ma.themtical Papers, vol. 1(1889), p. 127.

3. AawUR CAYLEY, Note on a system of imaginaries, The Collected MthematicM Papers,vol. 1(1889), p. 301.

4. H. S. M. COXETER, The polytopes with regular-prismatic vertex figures, Part 1, PhilosophicalTransactions of the Royal Society of London (A), vol. 229(1930), pp. 329-425.

5. H. S. M. COXEWER, The polytopes with regular-prismatic vertex figures, Part 2 Proceedingsof the London Mathematical Society (2), vol. 34(1932), pp. 126-189.

6. H. S. M. COXETER, The binary polyhedral groups, and other generalizations of the quaterniongroup, this Journal, vol. 7(1940), pp. 367-379.

7. H. S. M. COXETER, Regular Polytopes, London, 1947.8. L. E. DCKSON, On quaternions and their generalization and the history of the eight square

theorem, Annals of Mathematics (2), vol. 20(1919), pp. 155-171.9. L. E. DCKSON, A new simple lheory of hypercomplex integers, Journal de Muthmtiques

Pures et AppliquSes (9), vol. 2(1923), pp. 281-326.10. L. E. DICKSON, Algebras and their Arithmetics, Chicago, 1923.11. TttottoID GOSSET, On the regular and semi-regular figures in space of n dimensions, Mes-

senger of Mathematics, vol. 29(1900), pp. 43-48.12. W. R. HAMILTON, Note respecting the researches of John T. Graves, Esq., Transactions of

the Royal Irish Academy, vol. 21(1848), pp. 338-341.13. ADOLF HURWlTZ, ber die Zahlentheorie der Quaternionen, Mthematische Werke, vol. 2

(1933), pp. 303-330.14. JOHANNES KIRMSE, ber die Darstellbarkeit natiirlicher ganzer Zahlen als Summen yon acht

Quadraten und iber ein mit .diesem Problem zusammenhgngendes nichtkommutativesund nichtassoziatives Zahlensystem, Bericht fiber die Verhandlungen der SchsischenAkademie der Wissenschften zu Leipzig, vol. 76(1924), pp. 63-82.

15. KRT MAHLER, On ideals in the Cayley-Dickson algebra, Proceedings of the Royal IrishAcademy (A), vol. 48(1943), pp. 123-133.

16. EMILE MATttIEU, Sur une formule relative la thtorie des nombres, Journal ffir die reineund angewundte Mathematik, vol. 60(1862), pp. 351-356.

17. RuTtt MOUFANG, AlternativkSrper und der Satz vom vollst(indigen Vierseit (D), Abhand-lungen aus dera Muthematischen Seminar der Hamburgischen Universitt, vol. 9(1933), pp. 207-222.

578 . s. M. COXETER

18. RUTH MOUFANG, Zur Struktur von AlternativkSrpern, Mathematische Annalen, vol. 110(1934), pp. 416-430.

19. LVDWG SCHLXFL, Theorie der vielfachen Kontinuitt, Denkschriften der Schweizerischennaturforschenden Gesellschaft, vol. 38(1901), pp. 1-237.

20. ERNST WITT, Spiegelungsgruppen und Aufzgihlung halbeinfacher Liescher Ringe, Abhand-lungen aus dem Mathematischen Seminar der Hansischen Universittt, vol. 14(1941),pp. 289-322.

UNIVERSITY OF TORONTO.

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Integral Cayley numbers