A simple and relatively efficient triangulation of thencube
Discrete Comput Geom
A Simple and Relatively Efficient Triangulation of the nCube*
M a r k H a i m a n 0
0 Department of Mathematics, Massachusetts Institute of Technology , 77 Massachusetts Avenue,Cambridge, MA 02139 , USA
The only previously published triangulation of the ncube using o(n!) simplices, due to Sallee, uses O(n2n!) simplices. We point out a very simple method of achieving O(p"n!)simplices, where p < 1 is a constant.

and hence Sallee uses at least O(n2n!) simplices. (Readers of [
7
] should observe
that the function g(n, m) of that paper is actually A(n  1, m  1), a fact not noted
there.)
The construction given below triangulates I n with O(p"n!) simplices, where
p < 1. In fact, given a triangulation of I n into T(n) simplices for any particular
value of n, we can take p = (T(n)/n!) 1/n. The smallest value for p obtainable from
triangulations published to date is p = (13,248/40,320)1/a ~ 0.870 from Sallee's
triangulation of I s. We remark that Todd [
11
] proposed the quantity R(n) =
(T(n)/n!) 1In as a measure of a triangulation's efficiency. Previous constructions
have lim,~ ~ R(n) = 1, whereas our results show that any value of R(n) achievable
for one triangulation is achievable asymptotically. This is still far from R(n) = o(1),
let alone the lower bound of R(n) = O(n 1/2).
2. Construction
Definition. A polyhedral decomposition of an ndimensional polytope P is a union
P = T1 u T 2 u .  . w TR of ndimensional polytopes 7]i such that for all i,j the
vertices of T/are vertices of P and Ti n Tj is a (possibly empty) face of both Ti and
Tj. If each Ti is a simplex, then {Ti} is a triangulation of P.
L e m m a 1. Every polyhedral decomposition of P can be refined to a triangulation.
Proof. We require triangulations 0i of the T~ such that 0~ and 0j induce the same
triangulation on T / n Tj considered as a face of T/and as a face of Tj. N o w there are
wellknown constructions [
4
], [
6
], [
9
], [
12
] whereby we associate to a total
ordering ~ of the vertices of a polytope T a triangulation 0 in such a way that the
triangulation induced on each face F G T is the one associated to the restriction of
a to the vertices of F. Hence fixing any total ordering ~to of all the vertices of P and
triangulating each T~in accordance with ao we obtain compatible triangulations 0~
as required. []
L e m m a 2 [
1
]. Every triangulation o f A k x A t uses exactly (k +/)!/k! l! simplices,
where A, denotes an ndimensional simplex.
Proof. Realize A k in R k a s the convex hull of 0 and the unit coordinate vectors
e ~ = ( 0 . . . . . 0, 1,0 . . . . . 0). Likewise A~. Then A k A,___R k+t has vertices 0,
ei (I < i < k + / ) , and el + e~ (1 < i < k < j < k + / ) . Its volume is V(Ak)V(AI) =
1/k! l !. We claim every nondegenerate (k + / )  s i m p l e x A spanned by vertices of
A k A l has volume 1/(k +/)!. N o t e that there are affinelinear symmetries of
A k x At acting transitively on the vertices. These preserve volume, so we can
assume 0 is a vertex of A. Then _ (k + / ) ! v(A) is the determinant of the matrix M
whose rows are the coordinates of the other vertices; we are to show that this
determinant is _+1. If some el is a vertex of A, then expanding by minors on the
corresponding row gives the result by induction. If not, then M is the edgevertex
incidence matrix of a (k, /)bipartite graph with k + l edges. Having one too m a n y
edges to be a tree, this graph must contain a cycle, necessarily even. But then M is
singular, contrary to hypothesis. [ ]
Theorem 1. Given a triangulation {$1 . . . . . S~} of a kdimensional polytope P and a
triangulation { TI . . . . . Tt} of an ldimensional polytope Q, there exists a triangulation
of P Q using s . t . ( k + l)!/k! l! simplices.
Proof It is easy to see that {S, Tj} is a polyhedral decomposition of P Q.
Refine it to a triangulation by Lemma 1. Each S~ T~ will then contain
(k + l)!/k! 1! simplices by Lemma 2, establishing the result. [ ]
Corollary 1. I f I" can be triangulated into T(n) simplices, then I k" can be
trianyulated into [(kn)!/(n !)k] T(n)k = pk,(kn) ! simplices, where p = (T(n)/n !)1/,.
Proof
Immediate from Theorem 1 by induction on k. [ ] References
1. L.J. Billera , R. Cushman , and J. A. Sanders , The Stanley decomposition of the harmonic oscillator , Nederl. Akad. Wetensch. Proc. Ser. A 91 ( 1988 ), 375  393 .
2. R. W. Cottle , Minimal triangulations of the 4cube , Discrete Math. 40 ( 1982 ), 25  29 .
3. C. Lee , Triangulating the dcube, in Discrete Geometry and Convexity , J. E. Goodman , E. Lutwak , J. Malkevitch , and R. Pollack, eds., New York Academy of Sciences, New York ( 1985 ), pp. 205  21 I.
4. C. Lee , Some notes on triangulating polytopes, in Proc. 3 . Kolloquium iiber Diskrete Geometrie, Institut ffir Mathematik, Universit/it Salzburg ( 1985 ), pp. 173  181 .
5. P. S. Mara , Triangulations of the Cube, M. S. Thesis , Colorado State University ( 1972 ).
6. J. F. Sallee , A triangulation of the ncube , Discrete Math. 40 ( 1982 ), 81 86 .
7. J. F. Sallee , Middlecut triangulations of the ncube, S I A M J. Algebraic Discrete Methods 5 , no. 3 ( 1984 ), 407  419 .
8. W. D. Smith , Polytope triangulations in dspace, improving Hadamard's inequality and maximal volumes of regular polytopes in hyperbolic dspace. Manuscript , Princeton, NJ ( September 1987 ).
9. R. P. Stanley , Decompositions of rational convex polytopes , Ann. Discrete Math. 6 ( 1980 ), 333  342 .
10. R. P. Stanley , Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole, Monterey, CA ( 1986 ).
11. M. J. Todd , The Computation of Fixed Points and Applications, Lecture Notes in Economics and Mathematical Systems , Vol. 124 , SpringerVerlag, Berlin ( 1976 ).
12. B. Von Hohenbalken , How To Simplicially Partition a Polytope, Research Paper No. 79 17 , Department of Economics, University of Alberta, Edmonton ( 1979 ).