The Dali Geometric CrossAuthor(s): Paul ScottSource: Mathematics in School, Vol. 21, No. 4 (Sep., 1992), pp. 20-21Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214906 .

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G

E the Pil CROSS

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C by Paul Scott, University of Adelaide, South Australia

The illustrated painting, Crucifixion, or Corpus Hyper- cubicus is by the Spanish surrealist painter Salvadore Dali (1904-1989). It has special mathematical interest because of the geometric nature of the cross. The cross itself is made up of eight adjoining cubes. We investigate whether this configuration of cubes has any particular geometric significance.

Fig. 1 (a) (b) (c)

Metropolitan Museum of New York

We might compare the Dali cross with a more orthodox cross, the plane face of which comprises six adjoining squares (Fig. l(a)).

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Notice that we would in fact obtain this plane figure by looking at the Dali cross "straight on". Model makers will recognize this shape as the standard net for constructing a cube. If we now looked at this planar cross "edge on" we would see four adjoining line segments. These can be thought of as making up a "net" for a square (Fig. l(b), (c)).

The process appears to end here, for if we look at the four collinear line segments end on, we see a single point, and we can't make much out of that! However, the idea of dimension is clearly present here. Perhaps we might get more insight by considering the constructed figures (cube, square), rather than the nets.

If we take a unit square and translate it 1 unit in a direction perpendicular to itself, and think of the vertices of the square leaving "trace paths", we obtain a unit cube. Similarly, the square itself can be obtained by taking a line segment of length 1 and translating it 1 unit in a direction perpendicular to itself. In turn, the line segment is obtained by translating a point 1 unit (perpendicular to itself!). We thus obtain the sequence in Fig. 2.

Salvadore Dali began his formal art education at the San Fernando Art Academy in Madrid. He had a stormy career, being suspended for a year. Finally he was permanently expelled after publicly refusing to take an examination, declaring "None of the professors of the school of San Fernando being competent to judge me, I withdraw"

Mathematics in School, September 1992

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Fig. 2

Could it be that this sequence might be extended beyond the cube? Thus, could we translate the cube 1 unit in a direction "perpendicular to itself? This is a bit hard to visualize, but this is only because we live in a 3-dimensional world. Suppose we obtained a "4-dimensional" cube, or hypercube by this means. What properties would such an object have?

Let us draw up a table listing the numbers V, E, F,... of vertices, edges, faces, ... of the objects in the sequence. It is easy to fill in the first four columns (Table 1).

Table 1

Point Segment Square Cube

V 1 2 4 8 E 0 1 4 12 F 0 0 1 6

We now ask if there are any patterns amongst the entries in the table. Perhaps there is a rule akin to the rule for Pascal's triangle, which will allow us to deduce the entries from those which come before. From the given values, and keeping in mind the method of obtaining each figure from the one before, we find that there is indeed such a rule*, giving rise to a fifth column of values. Adding a further row under the heading Cell (C), denoting the number of "3 dimensional faces", we obtain Table 2.

Table 2

Point Segment Square Cube Hypercube

V 1 2 4 8 16 E 0 1 4 12 32 F 0 0 1 6 24 C 0 0 0 1 8

Notice that even if we don't believe in the existence of the hypercube, we can now say a lot about it: it has 16 vertices, 32 edges, 24 square faces, 8 cubical (3 dimensional) faces, and presumably 1 four-dimensional face.

Finally, let us try to relate this four-dimensional solid to the Dali cross. It is possible to represent a three- dimensional solid by a plane figure - we do this all the time with photographs and paintings. However, when we draw a picture of a cube say, some distortion takes place:

a

b a + 2b

Mathematics in School, September 1992

the square faces do not all remain square. For example, in the picture of a cube in Fig. 3, only two of the six square faces appear as squares. To obtain this cube from the net, identify the inner square above with the square of the net which is surrounded by four squares; the bottom square of the net is then folded up to complete the cube.

Fig. 3

In the same way, a hypercube can be pictured as one cube placed inside another, with the corresponding vertices joined: we can do this with a 3 dimensional model, or in 2 dimensions, as in Fig. 4.

Fig. 4

Here, the central cube is in fact surrounded by six other cubes, but these appear to be distorted because our "pic- ture" is in a lower dimension. There are thus a total of eight cubes as we expect from our table. We can now almost visualise how the hypercube is constructed from the net which makes up the Dali cross: the central cube is surrounded by six others as above, and the bottom cube is "folded up" to enclose the others.

The title of the painting shows that Dali was aware of the significance of his cross. Perhaps he painted it this way to give an extra "spiritual dimension".

For further investigation 1. Look at other Dali paintings: the graphic Soft construc- tion with boiled beans, the thought-provoking The persistence of memory, and the beautiful Christ of St John of the Cross. The Tate Gallery has published a catalogue of Dali's works.

2. Look for other sequences like point, segment, square, cube, .... For example, try point, segment, triangle, .... What is the rule of construction? Form a table listing the number of vertices, edges, faces, .... Are there number relationships here? What can you say about the four dimensional simplex?

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