Projective planes and the Tripartite Unity


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

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  
Figure 2

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, a, b and c in order to obtain the word and classify the represented compact surface.

(a)      (b)     (c)     (d)     (e)
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 5 shows a choice of another cut (dashed line), indexed by the letter d.

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

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-1dcc (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-1dcc 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

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

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



[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.