WALTER BENZ

CONTINUOUS

PROPORTION

RELATIONS

AND

FUNCTIONS Dedicated to HeImut Salzmann on the occasion of his 60th birthday

ABSTRACT. Continuous proportion functions in three or more dimensions are classical.

. Let Z" be the set of all n-dimensional boxes of R" (n ~> 2) and let ' ~ ' be an equivalence relation on E". Two boxes Q, Q'e z" are called reciprocal if they are homothetic and if there exists an edge of Q which is congruent to an edge of Q'. Reciprocal rectangles are important in architecture; examples can be found in many building designs since ancient times (see, among others, Le Corbusier [8]). The following theorem will be proved: T H E O R E M 1. Suppose that (Z", ,-~) satisfies: (i) Reciprocal boxes Q, Q' e E" are always equivalent, Q ~ Q'; (ii) Q~ ~ Q, Q'~ ~ Q', Q~ ,,~ Q'~ =~ Q ~ Q'. (In words: Let Q1, Q2. . . . -+Q and Q'I, Q'2. . . . o Q ' be convergent sequences of boxes Q~, Q'~~ E" with limit boxes Q, Q' e z" such that Q~ ~ Q'~ holds true for all positive integers v. This implies that also the limit boxes are equivalent.) Then for n >1 3 two homothetic boxes Q, Q' E E" must always be equivalent. We call (E", ~) a proportion relation if (i) is satisfied. We call a proportion relation continuous if also (ii) holds true. REMARKS. (1) One step of our proof of Theorem 1 depends on a result of Weyl [12] on equidistribution. (2) Theorem 1 is not true in the case n = 2. This phenomenon leads to interesting structures (Z 2, ~ ) which can be used by the architect for constructions of subdivisions of areas in pairwise equivalent rectangles. (Gronau [7] has developed computer graphics for special subdivisions of this type.) Another result of this note is a corollary of Theorem 1 and it concerns proportion functions. In our present definition of a proportion functionf we will replace the usual range off, namely [1, c~) (see Alsina and Benz [2]), by Geometriae Dedicata 36: 139-149, 1990. © 1990 Kluwer Academic Publishers. Printed in the Netherlands.

140

WALTER

BENZ

an arbitrary topological Hausdorff space X. This allows us to specialize X in different ways. Now let X be a topological Hausdorff space and let n be an integer greater than 1. A function f : (0, ~)" ~ X will be called an n-place proportion function if it satisfies (1) f ( x i , . . . , x , ) = f ( x , , ( l ) . . . . . x,(,)) for all (xl . . . . . x,)e(0, oo)" and all permutations cr of {1, 2, . . . , n}; (2) f ( x l , . . ., x,) = f ( ( x , / x l ) " x l , ( x . / x l ) " x2, .. ., ( x , / x l ) " x.) for all (xl . . . . .

x.)~(0, oo)". REMARK. Note that we do not require the normalization axiom f(x i,...,x.)=l

for a l l x l = x 2

.....

x,>0

for a proportion function. Continuity of f, however, implies f ( x . . . . . x) = const (see Theorem 2). Writing the real numbers x l , . . . , x, in their natural order, x'~ ~< x~ ~< ... ~< x',, we put Medi(xl . . . . . x,) := x'i+ 1 for i = 1. . . . . n - 2. where Med stands for 'medium'. If now g: {(Yl, .-., Y , - 1 ) e R ~ - l l 1 <~Yl <~ "'" <~Y,-1} ~ s is taken arbitrarily, then /Medlx Med._2 x M a x x ~ f(x):=O~-~nx .... ' Minx '~nnxJ must be a proportion function with x : = (xx . . . . . x,)e(0, oo)". Such proportion functions are called classical. There exist non-classical proportion functions for all n >/2. The second result is now T H E O R E M 2. Suppose that n >>,3. Then every continuous proportion f u n c t i o n f: (0, ~)" ~ X is classical. REMARKS. (1) A classical proportion function, such that the associated function g is continuous, is itself continuous. (2) All continuous proportion functionsf in the case n = 2 and X = [1, oo), which satisfy the normalization axiom, were determined by Benz [4] and Moszner and Nawolska [10]. They need not be classical.

CONTINUOUS

PROPORTION

RELATIONS

AND

FUNCTIONS

141

(3) Concerning the theory of proportions in architecture compare Ghyka [6], March and Steadman [9] and Scholfield [11], among others. A third result of this note, which is also a corollary of Theorem 1, is an answer for n > 2 to the following question: Suppose that (Z", ~ ) is a proportion relation. Find necessary and sufficient conditions for Q -,~ Q' to hold true iff Q, Q' are homothetic. T H E O R E M 3. Suppose that (Y,", ,-~), n ~> 3, is a proportion relation. Then the followin9 properties are equivalent: (a) Two boxes Q, Q' e Z" are homothetic iff Q ,,~ Q'; (b) (Z", ~ ) is continuous and sharp. We call a proportion relation sharp if the following holds true: (,)

Q ,-~ Q' and Min Q = 1 = Min Q' imply that Q, Q' are congruent.

Here Min Q denotes the minimal length of the edges of Q.

.

In this section Theorem 1 will be proved. L e t / 2 be the set I2:= {(x~. . . . . x , ) e ~ " t 0 < xl <~ x2 <~ "" ~ x,} and define

in such a way that x 1 ~ x 2 ~. " ' " Xn of q~(Q) = (xl, 1 2 , . . . , X n ) a r e the lengths of the edges of Q in their natural order, tp is obviously surjective. For x, y e/2 we write x ,-~ y if and only if there exist boxes Qo, Ro ~ E" such that tp(Qo) = x, q~(Ro) = Y, Qo ~ Ro. L E M M A 1. ',-~' is an equivalence relation o n / 2 . Suppose that x ,,~ y for x, y ~ Ig and that tp(Q) = x, ~o(R) = y. Then Q ~ R.

Proof. Because of x ~ y there exist Qo, Roe Z" with ~o(Qo) = x, ~o(Ro) = y, Qo "~ Ro. The boxes Q, Qo are thus congruent and hence reciprocal. This implies Q ~ Qo by (i). We also have R ~ R o. Hence Q ~ R because of Qo "-" Ro. L E M M A 2. Suppose that x = (xl . . . . . x , ) e /2. Then (-Xl~'...(x~-IY"-~'x

x

~

\TJ

0~1, . . . , O~n_ l ~ 7/.

/

forall

142

WALTER

BENZ

Proof Take i, j~{1 . . . . . n} and boxes Q, R with q~(Q)= x and (o(R) = (xi/xs). x. The boxes Q, R are reciprocal since xi occurs also as the length of an edge of R. Hence x ~ (xJxs). x. Applying this formula again for #, v ~ { 1. . . . . n} we get x ~ xj

(xJxj)x~ (x,/xj)x~

x, x= x~

x~ . x ~ . x. x j x~

Iterating this process finishes the proof. Denote by S" the surface

s":= {(~2, ,hC ,~2¢, ..., &-2~, 0 ~ / : 1 0 < ~ < 1} of R". For z : = (4 2, 'h~ . . . . , ¢)ES"

define (z):={p'zll
~":= U (:). zeS n

L E M M A 3. Let x be an element of L" such that xl < x,. Then there exists an element y E M" with x ~ y. Proof Observe that z := (xl/xZ)'x belongs to S". Denote as usual by [a] the largest integer not exceeding a. Put In x, ] k:= -[.lnx~n x.j and (Xl~ k'x2" and P : = \ x , / xl

y:= p.z.

Because of Lemma 2 we get x ~(xl~k'x= \X./

p ' z = y.

We finally prove ye(z), i.e. 1 < p <~x,/x 1. The definition of k leads to In xn

ln(xl/x.)

l<-k~<--

In x.

ln(xl/x,,)

CONTINUOUS

PROPORTION

RELATIONS AND FUNCTIONS

143

and hence to X1

(XI~ k

,

~<--,

Xn

k, X n f

Xn

--~<,

1

i.e.l
X, X1

An arbitrary point of S" can be written in the form

where0<~
>~wl~>w2~>...~>w,

2 ~> 0- T a k e such a point z and

~7):= ~mw,-rmw,~-l, Ze(Z) for all positive integers m. Because of

~z~) = ~-[,.wl](~w,),..~

Z

we get (7) ~ (1/~)z by applying L e m m a 2. L E M M A 4. Suppose that w 1 ~ [0, 1] is not rational and that p e (z) f o r a f i x e d point z = ~.(~, ~ ,

. . . . ~ . - 2 , 1)eS".

Then there exists a sequence ml, m2, . . . o f positive integers m~ such that ('~') ~ p. P r o o f Because of

p = p z = p ~ . ( ~ , ~ , . . . . ,1), ~ < p~ < 1,

we get p = U ' z / ~ for a suitable ~ e [0, 1). The sequence m w 1 - [row1] , m = 1, 2, 3, . . . , is equidistributed in [0, 1]. According to Weyl 1-12, Satz 2], there hence exists a sequence miw 1 [m~w~] ~ ~. Thus -

-

P"

L E M M A 5. Suppose that w 1 ~ [0, 1] is not rational and that z = ~" (~, ~ " , .. ., 1) ~ S". T h e n p, q ~ (z) implies p ~ q. P r o o f According to L e m m a 4 there exist sequences ('~') ~ p

and

(~;) --. q.

Because of (~') ~ (1/#)z ~ (}~) and of axiom (ii) we thus get p ~ q.

144 LEMMA

WALTER BENZ 6. L e t x = (xl . . . . .

x.) be an element o f L" such that x~ < x . and

that In x2 - In x . j' , _

in x~ - In x .

is not rational. Then x ~ t x f o r all t > O. P r o o f A c c o r d i n g to L e m m a 3 a n d its p r o o f we get "xlXkl

tx

\-~,/

• t x = : y' G M"

ka =

_r_ Llnx:

with - tnx.J

and

k2

=

-

I 'n'" l l n x l - lnx._l"

B e c a u s e of

x~

(tx,)

z:=--sd'X = "tx Xn (tXn) ~ we also h a v e y, y ' e (z). C a l c u l a t i n g w: f r o m X1 -i'x Xn

= z =:~-(¢

~w,, . . . , l i

we get X1

X2

,Z:

= xZ'

i.e. w 1 = r. H e n c e L e m m a 5 i m p l i e s y ,-, y' a n d t h u s x ~ y ~ y' ,,~ tx. L E M M A 7. L e t x = (xi, . . . , x,) be an element o f 12 such that x : < x2. Then

x ~ t x f o r all t > O. P r o o f N o t e t h a t xx < x2 ~< x.. B e c a u s e of L e m m a 6 we h e n c e m a y a s s u m e that In x2 - In x , In x l - In x , is r a t i o n a l . O b s e r v e t h a t ( x , / x l ) l/(vx/~- 1) >/1 for v = 1, 2, 3, . . . . H e n c e L"~:=

(

xl,...,x,-1,

L n~ t(~ --~ tx

"'\-~x]

]~x,

CONTINUOUS

PROPORTION

RELATIONS

145

AND FUNCTIONS

for v -0 oe. L e m m a 6 yields %) ~ t~ because lax2-

lnxl-

ln(xn'(xn~l/v~/2--1)~ \Xl/ / ln(x.('x-z~]l/v/'2-1)]: : " \ \xl/

1- r VN / ~

" /

is not rational for all v = 1, 2 . . . . .

(Note that r e [0, 1).) Hence (ii) implies

X~tX.

L E M M A 8. L e t x = (xl . . . . . x~) be an element o f 15, such that x I < x~. Then x ,,~ t x f o r all t > O. Proof. Because of L e m m a 7 we m a y assume xl = x2. N o w I~

X2~ X2~

I~ ~ t"

X3, . . . ~ X n

X2, X2, X3, ...,

~ X

Xn

~

tX

by (v/(x + 1))x 2 < x 2 and L e m m a 7. Finally observe (a, a . . . . . a) ~ t(a, a . . . . .

. . . . .

a) for all positive a and t:

. . . . .

a,.

We hence have x ~ t x for all t > 0 and all x e/2. T a k e now two homothetic boxes Q, Q ' ~ 2 ~. Put q~(Q) =: x. Then ~o(Q') = t x with a suitable t > 0. N o w x ~ t x and L e m m a 1 imply Q ,-- Q'. This finishes the p r o o f of T h e o r e m 1. R E M A R K S . (1) In order to give an example of a continuous p r o p o r t i o n relation (I~2, ~ ) for which T h e o r e m 1 is not true we refer to [4]: T a k e there f: D--* I belonging to the function of formula (17) in [4]. Call now two rectangles of sides xl, x2 and yl, Y2, respectively, equivalent iff(xl, x2) = f ( Y l , Y2). (2) It is possible to determine all p r o p o r t i o n relations of Rn: F o r y ~/2 define the track z(y) by 1

~(Yl . . . . . Y.): . . . . .

I

}

146

WALTER

BENZ

Since tracks are equal or disjoint we consider an arbitrary mapping 7: T ~ H, where T is the set of tracks of/2 and where H ¢ ~ denotes an arbitrary set. Take now Q, Q'~E ~ and put q~(Q)=:q, q~(Q') =:q'. The general proportion relation is now given as follows: Define Q ~ Q' iff ~(r(q)) = ~(z(q')).

.

In this section Theorem 2 will be proved. Starting with a continuous proportion function f : (0, ~)" ~ X we associate t o f a proportion relation (E n, ,-~). For Q, Q'~ E n define Q ,-~ Q' if

f(qg(Q)) = f(gv(Q')), where q~is the function of the beginning of Section 2. Then ' ~ ' is of course an equivalence relation. Let Q, Q' ~ E ~ be reciprocal boxes and put ~0(Q) = x = ( x l . . . . (p(Q')

=

X' =

t (XI,

, x~), ...

, X'n).

Then x' = tx and x} = xi for suitable i,j imply txj = x) = xi, i.e. x' = (xi/xj)x. But then we getf(x') =f((xi/xj)x ). For i = j we have Q --~ Q' by definition. In the case i # j we rearrange the components of x in such a way that the first component of the result y will be xj and the last one xl. Then axiom (2) of a proportion function leads to

f(y) = f (xl Y l , -xi- Y2, . . . , - -x, Yn) • \xj xj xj (Applying (1) twice gives us

f , x , =f,y)

and

f(x~. Y ) = f ( x ~ . x).

Hence

kX2 i.e.Q~Q'. We now show that the proportion relation (E ~, ~), associated to a continuous proportion function, is itself continuous. In fact, from Qv ~ Q, Q'v ~ Q' we get f(q~(Q,)) ~f(q~(Q)), f(cp(Q'O ~f(~o(Q')) since f is continuous. Assuming Q~ ,,~ Q', for v = 1, 2 . . . . leads tof(q~(Q,)) = f(q~(Q'0) for v = 1, 2 . . . . and hence to f(qg(Q)) =f(~o(Q')), i.e. Q ~ Q'.

CONTINUOUS

PROPORTION

RELATIONS

AND FUNCTIONS

Applying now Theorem 1 gives us f ( x ) = f ( t x ) x~(0, ~)". We write for x~(0, ~)"

147

for all t > 0 and all

f ( x l . . . . . x.) = f ( M i n x, Medl x, . . . , Med._2 x, Maxx). Hence

f(x)=f

Med i x Max x'~ 1, M i n x . . . . . ~ J

turns out to be classical by putting

9 ( Y l , . . . , Y , - 1 ) : = f ( 1, Ya. . . . . Y,-,) in the case 1 ~< y~ ~< ... ~< y,_l.

.

The only interesting cases for applications are of course n = 2 and n = 3. If one accepts that continuity is a reasonable assumption in many practical situations like the fact that empirical curves are very often approximated by continuous curves, then Theorem 1 says that proportion relations in three dimensions are less fascinating than those in two dimensions. Theorem 2 expresses the same for proportion functions. We now define the range X"- 1 of an n-place proportion function f by X " - I : = {(y~. . . . . y , - ~ ) ~ " - ~ [ 1 ~< y~ ~< ... ~< y,-1} with the induced topology of N"-1. The following are in some sense the easiest examples for ne {2, 3}:

f(x, y) -

Max(x, y) Sin(x, y)

and

(Meal(x, y, z) Max(x, y, z)~ Y, z)' Sin(x, y, z))"

f(x, y, z) = \ ~

The proportion relation associated to (Med(x, y, z) Max(x, y, z))

f(x, y, z) = \ S i n ( x , y, z)' Sin(x, y, ~is sharp. Now we proceed to the proof of Theorem 3. It is trivial that (a) implies (b). Assume that (b) holds true. According to Theorem 1 we only have to prove that (a') Q ~ Q' implies that Q, Q' are homothetic.

148

WALTER BENZ

Let Q, Q' be equivalent boxes of Z". P u t t i n g x : = ~o(Q) a n d x' : = cp(Q') we get x ~ x'. A c c o r d i n g to the end of Section 2 we have z ,,~ t z for all t > 0 a n d z ~ L n. H e n c e

1 1 M i n x ' X ~ x ~ x' ~ M i n x ' "x'. T a k e boxes R, R ' e E" with q~(R) =

1 Minx

"x

and

~0(R') =

1 --'x'. Minx'

L e m m a 1 then implies R ~ R'. Since M i n R = 1 = M i n R' we get t h a t R, R' are congruent. Hence 1 Minx

x

_

1 --'x, M i n x'

t

i.e.

X p

Minx' =--'x. Minx

This implies t h a t the boxes Q, Q' are h o m o t h e t i c . R E M A R K S . (1) T h e o r e m 3 is n o t true in the case n = 2. (2) Call two p r o p o r t i o n functions f/: (0, ~ ) " ~ X "-1, n>~2, i = 1, 2, essentially equal, f l ( = ) f 2 , if they induce the same p r o p o r t i o n relation. Then, for n >~ 3, (Mea t x

Maxx')

f ( x ) ( = ) \. M~-nx . . . . ' M i n x /

p r o v i d e d t h a t f is c o n t i n u o u s a n d sharp: f ( x ) = f ( x ' ) M i n x = 1 = M i n x ' implies x = x'.

with x, x ' e I 2

and

REFERENCES 1. Acz61,J., Lectures on functional equations and their applications, Academic Press, New York, London, 1966. 2. Alsina, C. and Benz, W., 'Proportion functions in three dimensions', Aequat. Math. 37 (1989), 293-305. 3. Alsina, C. and Trillas, E., Lecciones de Algebra y Geometria (Curso para estudiantes de Arquitectura), Ed. Gustavo Gili, S.A., Barcelona, 1984. 4. Benz, W., 'A functional equations problem in architecture', Arch. Math. 47 (1986), 165 181. 5. Benz, W., ~.sthetische Rechtecksmal3e,die Mal3stabsunabh~ingigsind', Math. Methods Appl. Sci. 9 (1987), 53-58. 6. Ghyka, M. C., Est~tica de las proporciones en la naturaleza yen las artes, Ed. Poseid6n, S.L., Barcelona, 1977. 7. Gronau, D., Remark 4 (p. 99) in Report of Meeting, the Twenty-sixth International Symposium on Functional Equations, April 24-May 3, 1988, Sant Feliu de Guixols, Catalonia, Spain, Aequat. Math. 37 (1989), 57-127.

C O N T I N U O U S P R O P O R T I O N R E L A T I O N S AND F U N C T I O N S

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8. Le Corbusier, C. 1~., El Modulor, Ed. Poseiddn, S.L., Barcelona, 1976. 9. March, L. and Steadman, P., 'The geometry of environment', RIBA Publ., London, 1971. 10. Moszner, Z. and Nawolska, B., 'On the rectangle proportion', Wy~. Szkola Ped. Krakow., Rocznik Nauk.-Dydakt. Prate Mat. 12 (1987), 115-127. 11. Scholfield, P. H., The Theory of Proportion in Architecture, Ed. Labor, S.A., Barcelona, 1972. 12. Weyl, H., 'Ober die Gleichverteilung von Zahlen mod. Eins.' Math. Ann. 77 (1916), 313-352.

Author's address:

Walter Benz, Universitfit Hamburg, Mathematisches Seminar, Bundesstral3e 55, D-2000 Hamburg 13, F.R.G. (Received, August 15, 1989)