by

Ton Marar

Departamento de matemática aplicada e estatística – ICMC

Universidade de São Paulo, Brazil

The Swiss architect/designer Max Bill is known as an exponent of the so-called concrete art. His influence in modern art and his desire for an art based on mathematical concepts is remarkable [MB].

Bill’s metal sculpture entitled Tripartite Unity (Fig. 1 (a), (b)) won the first prize of the first Bienal de São Paulo – Brazil in 1951. Up to now, even the name of the sculpture was a mystery.

(a)
(b)

Figure 1

So, let us perform the identification of the surface represented
by the sculpture. Firstly we draw the surface (Fig. 3(a)) and its hidden
lines (Fig. 3(b)). Then we choose some cuts (Fig. 3(c)) in order to flatten
the surface. The three cuts chosen splits the surface into two pieces (Fig.
3(d)). The top piece is easily deformed into a flat portion (Fig. 3(e)).
Also we identify the edges created by the cuts with arrows and letters,
Figure 1

However, by considering the compact surface represented
by the sculpture, we show that it is topologically equivalent to the connected
sum of three projective planes with a point removed, corresponding to
the only boundary component of the represented surface.

To prove this, we cut the surface into pieces in order to obtain a plane model and then we read its word and hence identify the compact surface represented.

The reader can find a brilliant account on plane models and words of compact surfaces without boundary (closed surfaces) in Massey’s book [WM], see also Carter’s book [SC]. For surfaces with boundary, we consider the word of the surface obtained by gluing discs along the boundary components. For instance, the Möbius band and the projective plane have the same associated word, that is,**aa** (Fig. 2).

Figure 2

To prove this, we cut the surface into pieces in order to obtain a plane model and then we read its word and hence identify the compact surface represented.

The reader can find a brilliant account on plane models and words of compact surfaces without boundary (closed surfaces) in Massey’s book [WM], see also Carter’s book [SC]. For surfaces with boundary, we consider the word of the surface obtained by gluing discs along the boundary components. For instance, the Möbius band and the projective plane have the same associated word, that is,

Figure 2

(a)
(b)
(c)
(d)
(e)

Figure 3

Figure 3

Figure 4 shows the flattening of the top part. It
is a triangle with edges **a**, **b** and **c** and the white arcs
correspond to a portion of the boundary of the compact surface represented
by the sculpture. To flatten the bottom part we still need to perform another
cut.

Figure 4

Figure 4

Figure 5 shows a choice of another cut (dashed line),
indexed by the letter **d**.

Figure 5

**Acknowledgements:** Thanks are due to Angélica
de Moraes (art critic), Juan J. Nuño-Ballesteros (mathematician)
and Murillo Marx (architect) for their encouragement.

**References:**

[MB] BILL, Max. The mathematical approach in contemporary art. Arts and Architecture, Los Angeles, n. 8, 1954.

[SC] CARTER, J. Scott. How surfaces intersect in space. 2nd edition. New Jersey: World Scientific, 1995. (Series on Knots and Everything, v. 2).

[GF] FRANCIS, George. A topological picturebook. New York: Springer-Verlag, 1987.

[WM] MASSEY, William. Algebraic Topology, an introduction. Harcourt Brace & World Inc., 1967.

Figure 5

Figure 6 shows the bottom part flattened into a pentagon
with edges indexed by the letters **a**, **b**, **c** and **d**,
and the white arcs corresponding to the rest of the boundary component
of the represented compact surface.

Figure 6

Figure 6

By identifying the two flattened pieces by, say, the
edge **b** (Fig. 7) we end up with a plane model, another representation
of the compact surface represented by the Tripartite Unity. The word associated
to this plane model is **ada**^{-1}dcc (Fig.
8).

There are many operations that can be performed on the word of a compact surface that still keeps the topological type of the represented surface unchanged. One of such operations is the following: a letter in between two copies of another letter can be removed by changing its exponent. So, by means of this operation the word**ada**^{-1}dcc
represents the same compact surface as **aaddcc**. Also, the word associated
to the connected sum of two surfaces is the concatenation of the two words
of each summand. Hence, **aaddcc** represents the connected sum of
three projective planes. The single boundary component of the surface
represented by the Tripartite Unity is obtained by removing a point of
the sum of the three projective planes.

Figure 7 Figure 8

There are many operations that can be performed on the word of a compact surface that still keeps the topological type of the represented surface unchanged. One of such operations is the following: a letter in between two copies of another letter can be removed by changing its exponent. So, by means of this operation the word

Figure 7 Figure 8

The reader can find in Francis’ book [GF] the
compact surface depicted in Figure 9(a). Performing the two cuts indicated
by dashed lines (Fig. 9(b)) the surface is splitted into three Moebius
bands (Fig. 9(c)). Hence Figure 9(a) is topologically equivalent to the
Tripartite Unity. We call the sequence Fig. 9(a), (b) and (c) Three Variations
on the theme of the Tripartite Unity, with apologies to Max Bill.

(a) (b) (c)

Figure 9

(a) (b) (c)

Figure 9

[MB] BILL, Max. The mathematical approach in contemporary art. Arts and Architecture, Los Angeles, n. 8, 1954.

[SC] CARTER, J. Scott. How surfaces intersect in space. 2nd edition. New Jersey: World Scientific, 1995. (Series on Knots and Everything, v. 2).

[GF] FRANCIS, George. A topological picturebook. New York: Springer-Verlag, 1987.

[WM] MASSEY, William. Algebraic Topology, an introduction. Harcourt Brace & World Inc., 1967.