planes and the Tripartite Unity
Departamento de matemática aplicada e estatística –
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.
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.
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
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 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
Figure 5 shows a choice of another cut (dashed line),
indexed by the letter d.
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.
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.
[GF] FRANCIS, George. A topological picturebook. New York: Springer-Verlag,
[WM] MASSEY, William. Algebraic Topology, an introduction. Harcourt
Brace & World Inc., 1967.