48 I. J . SCHOENBERG

REGULAR SIMPLICES AND QUADRATIC FORMS

I. J . SCHOENBERG*.

1. R. E. A. C. Paley has investigated a certain type of orthogonalmatrices which he called [/-matrices (Paley [6]). The matrix

\ \ a i k \ \ {i, k = 0 , l , . . . , n )

is called a [/-matrix if aik = ± 1 and if any two rows (and therefore any twocolumns) are orthogonal to each other. Apart from the trivial cases7i = 0 or 1, [/-matrices of order n-\-l can exist only if n-\-l is amultiple of 4. After preliminary work by Sylvester, Hadamard ([4])>Scarpis, and Gilmanf, Paley has indicated certain classes of numbers n-\-1for which there exist [/-matrices of order n-\-\\ these classes cover, inparticular, all multiples of 4 not exceeding 200, with the exception of sixsuch multiples. The analytical interest of [/-matrices, apart from theircombinatorial and geometrical aspects discussed by J. A. Todd andH. S. M. Coxeter, lies chiefly in the fact, discovered by Hadamard, thatthere exist [/-matrices of order n-\-1 if and only if

max I det 11 xik \\ on =

in which case this maximum is reached for a [/-matrix ||z,-J| = II aik\\.I I CA> II II tfu I I

Let Un+1 = \\aik || be a [/-matrix. By obvious changes of signs in rowswe may assume that ai0 = + 1 . Denote by En the ^-dimensional Euclideanspace referred to rectangular coordinates. Coxeter ([2]) has noticed thereadily established fact that the n-\-1 points of En,

(1) P{= {aa, ai2) ..., ain) {i = 0, 1, ..., n),

are the vertices of a regular simplex an {i.e. all distances PiPk areequal) which is, of course, inscribed in the hyper-cube yn of vertices(± 1, • • •» db 1 ) a n ^ conversely, if (1) are the vertices of a regular an inscribedin yn (ornCyn), then ||%||0

?l5 with ai0= + 1 , is a [/-matrix of order TO+1.

Thus the problem of the existence of [/-matrices of order n-\-1 is equivalentto the problem of finding the dimensions n for which a an may be inscribedin yn-

* Received 10 August, 1936; read 12 November, 1936.f For references to the work of Sylvester, Scarpis, and Gilman, see Paley [6].

REGULAB SIMPLICES AND QUADRATIC FOK-MS.' 49

2. The present note does not contribute to a solution of this difficultproblem. It is concerned with the following less restrictive question.Denote by Ln the n-dimensional lattice of points of En with integralcoordinates. For what dimensions n can a an be inscribed in Ln% Thusinstead of requiring, as in the problem of Coxeter and Paley, that thevertices of an have coordinates ±1, we merely require these coordinatesto be rational integers*. A complete answer to this question is given bythe following theorem.

THEOBEM 1. It is possible to inscnbe a regular simplex an in theordinary rectangular lattice Ln only in the following cases:

1. If nis even, the construction is possible if and only if n-\-\ is a perfectsquare, i.e. for n = kh(h-\-l) {h=\, 2, 3, ...).

2. If n = 3 (mod 4), the construction is always possible.

3. If n=l (mod 4), the construction is possible if and only if n-\-\ is notdivisible to an odd exponent by a prime number of the form 4&+3, i.e. for

n= (2^+l ) 2 +(2 /c+l ) 2 - l (h, Jfc = 0, 1, 2, ...)t>

The construction is therefore possible for

n= 1, 3, 7, 8, 9, 11, 15, 17, 19, 23, 24, 25, 27, 31, 33, ...,

and impossible for the dimensions omitted. The yet unproved conjectureof Paley is that an(Zyn is always possible whenever 4 |w+l3 *.e.

n~ 3 (mod 4).

The second part of our theorem ensures the possibility of an c Ln for thesevalues of n.

3. Concluding this introduction, I should like to point out that theproblem an C Ln is to a certain extent the only non-trivial question of thiskind. Note first that an C Ln+1 is always possible, for the end points of thecoordinate unit-vectors of Ln+1 are the vertices of such a an. Considernext a regular polygon 11^ = Mx M2... Mn of n sides and let us ask in what

* For the elementary problem of disproving the possibility of <r2 c L2, see P61ya andSzego [7], 156; during its discussion in a problem-seminar at Swarthmore College, one ofmy students, Mr. Alan Bloch, asked whether cr2 c L3 is possible. His question gave theinitial impulse to the present investigation.

t This parametric representation follows from the remark that the numbers heredescribed are precisely the numbers which are the sum of two squares.

JOXTR. 45. E

50 I. J . SCHOENBERG

general (oblique) lattice L of m dimensions it might be inscribed. IfUn c L, we may obviously assume (since 0 has rational coordinates in L)

that the centre 0 of IIn is also a point of L. Let ON = 0Mx+0Mz.Notice that NczL and ONjOM2 = 2 cos (2ir/n). Hence cos (2n/n) must berational, which is the case only for n = 3, 4, and 6. We have thus proved:No regular polygon, except the triangle, square, and hexagon, can be inscribedin any general lattice of any number of dimensions. This implies similarnegative results for all the regular polyhedra and polytopes containingregular pentagons.

I. The connection with the theory of quadratic forms and the case in whichn is even.

4. Let a = Po Px ... Pn be a simplex in En and let aik = Pt- Pk {au = 0)be the lengths of its edges. Denote by L(a) the w-dimensional lattice of

points generated by linear combinations of the vectors p,- = Po P,- withintegral coefficients. The connection between the lattice L(o) and thepositive definite quadratic form

(2) F{x) = \ S {a^i-\ra^k-a%)xixk= S

is simple and well-known*. To a different S3̂ stem of vectors likepls ..., pn, which generate the same lattice L(a), there correspond? aquadratic form which is arithmetically equivalent to F, i.e. derivable fromF by a linear transformation with integral coefficients and determinant i 1,and conversely. A given positive definite quadratic form thus completelydetermines a corresponding lattice apart from a rigid displacement anda reflexion in a hyperplane, that is, determines its size.

In addition to a, let us now consider a second given simplexa' = P o ' . . . Pn' of En generating a lattice L{a'), and let F'(y1} ..., yn) beits corresponding quadratic form. When can a certain convenientmultiple of L(a), say XL(a) [obtained from L(a) by a similitude followed,perhaps, by a reflexion], be inscribed in i-(a')?

* See H. Minkowski [5], 243-254, where also references to Gauss and Dirichlet areto be found. The author first noticed the identity (2) in an article of Fre"chet [3], andrecognized its importance for certain problems of metric geometry (Schoenberg [8]).

REGULAR SIMPLICES AND QUADRATIC FORMS. 51

We need the following definitions. Two quadratic forms G(x) and"G'{y) are said to be rationally equivalent if there exists a non-singular lineartransformation with rational coefficients (y) = R(x), such that G(x) = O'(y).The forms O(x) and G'(y) are said to be rationally equivalent save for afactor if, for a suitable constant A, XG and G' are rationally equivalent.On the basis of the correspondence between lattices and quadratic formsthe following lemma, whose proof we may omit, is readily established.

LEMMA 1. Let a = P0P1 ... Pn and a' = P o ' P / ... Pn' be two given•n-dimensional simplices, and let

F(x)= S ( , o K *i,k=l \ / i,k=l

•be their corresponding quadratic forms. Let L(a) and L(a') be the lattices

•generated by the sets of vectors {Po P,} and {Po' P/} respectively. A suitablemultiple AL(a) of the lattice L(a) can be inscribed in L(a') if and only if.the quadratic forms F(x) and F'(y) are rationally equivalent up to a factor*.

5. The particular case with which we are concerned in Theorem 1 is aregular simplex a = an with all its edges equal to \/2, say, while a' = an'is made up of n mutually orthogonal unit-vectors with common origin.In view of (2), the corresponding forms are

,'(3) F(x) = 2 S x?+ S XiXk, F'(y) = yi*+...+yJ.i = l i¥=k

Now F{x) may be replaced for our purposes by a considerably simplerform. Indeed, the identity

(4) 2 S x?+ S xfxk= S *4^ (^+^i+^T+-) ( ^ = 0 if * >n)

i = l i¥=k i = l % \ l \ i l~TL I

;shows that F(x) is rationally equivalent to the form

(5) /„(*)= S i(i+l)x?= i atxf.i=l i = l

•Our problem is therefore to investigate the values of n for which an identity

1=1

* Minkowski does not seem to have pointed out this geometrical interpretation of.rational equivalence save for a factor, although it probably was known to him.

E2

52 I. J . SCHOENBERG

is a consequence of a certain linear transformation

(7) (y) = B(x) (|JR|#O)

with rational coefficients. Notice that A is necessarily rational.

5. Let now n be even. From (6) and (7) we have

Since A is rational and n is even, we conclude that n-\-\is a perfect square.In order to prove also the direct statement of the first part of Theorem 1we perform the following simple construction.

—>Let a' = OA1A2...An be the simplex generating Ln, i.e. let OA{ be

mutually orthogonal unit-vectors, and consider the regular simplexa = A0A1... An having with a a common base, with Ao and 0 on oppositesides of this base. The coordinates of Ao are readily found. Thecoordinates of the centroid C of the base A±A2 ... An are all equal to 1/n.Since the altitude of a regular simplex of side 1 is (n-\-l)^j(2n)^, we haveA0C=2^(n+l)^{2n)^ = (n+lfin-*. The points 0, G, Ao are collinearand the direction cosines of their line are all equal to n~$; hence thecoordinates of Ao are all equal to

nJ _ _ L (n 's/n v \ n ) n

This is a rational number if n+1 is a square, and then the regular simplexno can be inscribed in Ln. This proves that it is always possible to inscribea regular simplex in Lnifn-\-lis a perfect square. This completes the proofof the first part of Theorem 1.

II. Application of Minkowski's theory of rational equivalence of quadraticforms for the case of n odd.

7. Minkowski has set up a complete set of arithmetical invariants ofquadratic forms with rational coefficients with respect to transformationsof type (7) *; foremost among them is a certain sequence of units Gp(=±l)defined for all odd primes p. We start by determining these units of thequadratic form (5).

We make use of the Gauss sums

(8) <f>{h; n)=

* H. Minkowski, " Uber die Bedingungen, unter welchen zwei quadratische Formenmit rationalen Koeffizienten ineinander rational transformiert werden konnen" [5],219-239. See also the work of H. Hasse in the Journal jiir Math., 152 (1923), 129-148.

205-224.

REGULAR SIMPLICBS AND QUADRATIC FORMS. 53

recalling that

(8') <f>(h; n) = Vn(Mn)imn-1)]2 if (h, n)=l, n odd,

and

(8") <f>(h;n) = d<f>{h/d; n/d) if (h, n) = d,

where (h\n) denotes the Legendre-Jacobi symbol*.

The determination of Minkowski's units Cp depends on a wide generali-zation of the formula (8'). Let f(xx, ..., xn) be a non-degenerate formwith integral coefficients and determinant A, and let

f(h; N) = I e2*ihf(x1,...,xj/N [(hf JV) = i ] ,Xi, ...,Xn = l

If p is an odd prime and p^ | Af, Minkowski's extension of (8') (loc. cit.,page 229) is as follows:

(M) f(h; pl) = {h | p*+nt) Gp

Here cp stand for a second sequence of units i 1 and (M) is valid for suffi-ciently large values of t (e.g. t^d will do). If#>-[-A (d = 0), then Cp = + 1 ,as is seen from (M) for h = 1, t — 0.

The determinant of (5) is A = (n\Y(n-\-\); denote by A its largestsquarefree divisor, so that A is the squarefree part of n-\-\. Finally, let

#3|A, p&»\av{=v{v+l)}, p*\A [28,, = 3, S = 0 or 1, 3 = 8 (mod 2)].

If p-\-&, we know that Cp = + 1 ; if p | A (d > 0), apply Minkowski's formula(M) for h = 1 and t even and greater than d. We get

f{l-,pt)= S e2«i(aixr-+*)/ f[ lU=X

n

hence

(9) Op = i*nw-'-w-Mr<*>-w ( n\S odd

* See P. Bachmann [1], 165.•f pa | A indicates that pa is the highest power of p (3 > 0) which divides A.

54 I. J . SCHOENBEEG

To abbreviate, let

(V, odd

If p = 1 (mod 4), (9) gives

(11) Cp={n; !>}•

If p = —1 (mod 4), the exponent of* in (9) is modulo 4 congruent to

where a = E 1, with pa-\v (v= 1, 2, ..., n). In both cases (9) reduces toavodd

(12) 0 p = ( - l ) ' t o - « { » ; *>},

where o- stands for the number of those numbers v ̂ n which are divisibleby p to an odd exponent.

Let us now find a convenient expression for {n; p). If p -f A (8 = 0),we get, from (10),

{»;!>}=( n {{v-l)vp-*>v{v+l)p-°"}\p)\av Odd /

= ( n (v*-i)\p) = (-\\py= (-iyi(p-».\av Odd /

If, however, p\A{S—l, n=— 1 (mod p)}, then we have in {n \p) one morefactor, namely

{n{n+l)p-*»\p}= (nAp-*\p)= {-l\p){Ap-*\p)= (-l)U*

In both cases we therefore have

(13) {n; p}= (-iyU

Combining (12) and (13), we get

(14) Cp= {-

and this formula is valid for all odd primes p.

8. Let a be a positive squarefree integer. Our problem is to find whenthe quadratic form

aL' (x) = axi2+ax22+.. .+axn

2 (n odd)

is rationally equivalent to the form (5). One of Minkowski's trivialinvariants of a form is the squarefree part of its determinant; hence necess-

REGULAR SIMPLICES AND QUADRATIC FORMS. 55

arily a = A. Let us now compute the units Cp of the form Afn'. Wehave Cp = + 1 if p -f A. If p \ A, we apply Minkowsld's formula (M) again,with h = 1, t — 2, (d' = n), and obtain

/ ' ( I ; ^2) = {

whence Cy = i»{Kp-D} •-{*(»»—i

If p = 1 (mod 4) this gives Cp' = (Ap-1 \p). If p = 3 (mod 4) we have like-wise

Cpf=in-1(Ap-1\p) = (-

In each case we have

(15) Cp'= (—l)s^

for all odd primes p.In order to see for what values of n the units Cp and Cp are equal for all

p, let us form their product; (14), (15) give

CpCp'= (-l)«K*-l>K»+i>.

Now, if 7i= 3 (mod 4), we always have Cp Cp = + 1 , i.e. Op = Cp , andtherefore/7l and Afn' are rationally equivalent by Minkowski's fundamentalTheorem 1 (loc. cit., p. 222). This proves the second part of Theorem 1.If, however, n= 1 (mod 4), i.e. (n-\-\)f2 odd, we shall have CPCP = + 1for all odd primes p, if and only if S = 0, whenever p = 3 (mod 4). Thisproves the last part of our theorem.

References.

1. P. Bachmann, Zahlentheorie, 2 (Leipzig, 1894).2. H. S. M. Coxoter, "Regular compound polytopes in more than four dimensions",

Journal of Math, and Phys., 12 (1933), 334-345.3. M. Fr£chet, " Sur la definition axiomatique d'une classe d'espaces . . .", Annals of

Math. (2), 36 (1935), 705-718.4. J. Hadamard, " Resolution d'une question relative aux determinants", Bulletin des

Sciences Math. (2), 17 (1893), 240-246.5. H. Minkowski, Gesammelte Abhandlungen, 1 (Leipzig und Berlin, 1911).6. R. E. A. C. Paley, "On orthogonal matrices", Journal of Math, and Phys., 12 (1933),

311-320.7. G. P61ya und G. Szego, Avfgaben aus der Analysis, 2 (Berlin, 1925).8. I. J. Schoenberg, "Remarks to Maurice Freshet's article . . .", Annals of Math. (2), 30

(1935), 724-732.

Colby College,Waterville, Maine.