Modern computer graphics has emphasized the fact that
mathematics is very much a visual subject. Thus the visual mind and visual
thinking are of basic importance in mathematics just as in art. Furthermore,
visual thinking leads to seeing that mathematical forms can also generate
art forms as in the work of the Swiss artist Max Bill. In
Figure 1 is shown the granite sculpture
The Dutch artist Maurice Escher also applied mathematical
ideas to generate his art works. Recently the American artist Richard Serra
applied the idea of a rotating ellipse as the basis of his now famous at www.isama.org
provides links to a variety of interesting websites relating art, mathematics,
architecture, and mathematical visualization. For example, included in
the Directory under Directory are links to
the works of Bruce Beasley, Brent Collins, Helaman Ferguson, Robert Longhurst,
Charles Perry, John Robinson, and Keizo Ushio, as well as many others.
An excellent general reference on art and mathematics is the book Sculpture by Michele Emmer, MIT Press [1].
The
Visual Mind
One can say that the operative word that unifies art and
mathematics is SEEING. More precisely, art and mathematics are both about
SEEING RELATIONSHIPS. One can see certain mathematical forms as art forms
and creativity is about seeing from a new viewpoint. Thus it is all about
seeing. As the Basque sculptor Eduardo Chillida states "to look is one
thing, to see is another thing", "to look is to try to see", "to see is
very difficult, normally" [2]. I would like to add that
from my own experience as a research mathematician and sculptor, it can
take a lot of looking before one finally sees what has been there all the
time. Seeing better is a lifetime endeavor. An excellent related article
is We will now discuss a more complete way of seeing a three-dimensional
object that is called x-ray seeing, was known to the Cubist painters such
as Braque, Duchamp, and Picasso, as discussed in [4].
In particular, Cubists were led to showing multiple views of an object
in the same painting. In mathematics four-dimensional space is referred
to as and I refer
to seeing in hyperspace as hyperspace
([5]-[8]). Thus in hyperspace one could
hypersee a three-dimensional object completely from one viewpoint.
hyperseeing
Cubist painters approximated hyperseeing by showing multiple
views in the same painting. Hyperseeing can also be facilitated in our
three-dimensional world by viewing a hypersculpture defined as follows.
First, a sculpture is defined as an object in a given orientation relative
to a fixed horizontal plane (the base). Two sculptures are said to be
is a set of related sculptures. Thus viewing a hypersculpture allows one
to see multiple views of an object from one viewpoint, which therefore
helps to develop a type of hyperseeing in our three-dimensional world.
Furthermore, a hypersculpture is a more complete presentation of the sculptural
content of a three-dimensional form. A first hypersculpture called Rashomon
by Charles Ginnever is shown in Figure 2, where three
related sculptures are shown. Three other related sculptures are shown
below. Actually the form in hypersculpture has fifteen stable positions.
A second hypersculpture Rashomon by Arthur Silverman is
shown in Figure 3. Attitudes consists of
six related sculptures. The commen form in Attitudes consists
of a rectangle, parallelogram, and two triangles.
Attitudes
,
Arthur Silverman, 1996
Knots are usually presented by two-dimensional knot diagrams
that indicate where the knot crosses itself. Three diagrams of a trefoil
knot are shown in Figure 4.
Knots are usually presented as two-dimensional knot diagrams.
However, a knot is actually a three-dimensional object and a knot really
comes alive in a three-dimensional model of the knot which can be constructed
from wire, folded aluminum foil, folded aluminum foil with a wire insert,
copper tubing, or other material. Models of knots are ideal mathematical
forms that can generate ideas for sculptures. Examples of models of knots
made from folded aluminum foil are shown in Figure 5.
A knot made from copper tubing is shown in Figure 6.
Models of knots are perfect mathematical forms for generating ideas for sculptures. They are completely three-dimensional with no preferred top, bottom, front, or back. Furthermore, a knot can look completely different when viewed from different directions. In general, we can regard hyperseeing in our three-dimensional world as a more complete all-around seeing from multiple viewpoints. Since knots can look so different from different viewpoints, knots are excellent examples of interesting forms on which to practice hyperseeing. Knots are also open forms that one can actually see through. This is another reason that knots are ideal forms for hyperseeing. For example, if one makes a wire model of a trefoil knot with front view as in Figure4(a), then the top view will appear similar to Figure 4(b). A right side view of the model with a little tip forward will appear similar to Figure 4(c). In general, a model of a knot yields an infinite number of diagrams of the knot depending on your viewpoint of the knot. An interesting property that results from the open structure of a knot concerns looking at a model of a knot from opposite directions. Suppose we regard one viewpoint as the front viewpoint. The viewpoint directly opposite will be the rear view. In general, the rear view of an object cannot be determined at all from the front view. However, for a knot, the rear view will be the reflection of the front view with the crossings reversed.. For example, if one considers the views in Figure 4 as front views, then the corresponding rear views can be obtained by holding the turned page to the light and reversing the crossings. Another property of knots is that one sees all points
on a knot in any one view except for a finite number of crossing points
If one dips a wire model of a knot in a solution of liquid
soap and water, one obtains a soapfilm minimal surface with the knot as
the single edge of the surface. For example, the edge of a half-twist Möbius
band is a simple loop as shown in Figure 7(a). If a wire
is bent in this shape and dipped in a soap solution, one obtains a minimal
surface with a central disk. If this disk is puntured, then one obtains
the Möbius band minimal surface in Figure 7(b).
The corresponding soapfilm minimal surfaces for the trefoil
knots in Figure 4 are shown in Figure 8. In (a)
we have a triple twist Möbius band with one side. In (b)
we have a two-sided minimal surface. In (c) we have a one-sided
surface.
The exercise students of all ages really enjoy is dipping
wire models of knots in order to obtain the corresponding soapfilm minimal
surfaces. In order that students can learn to anticipate the shape of the
surface, it is helpful to use masking tape in order to approximate the
surface. An example is shown in Figure 9 for a trefoil
knot.
In Figure 8(c) we have a minimal surface
that could be described as a form consisting of three leaves with a central
space. We will now modify the knot so that the leaves become space and
the center is form. The modified knot is shown in Figure
10(a) and the corresponding minimal surface is shown in (b).
We refer to the knot in (a) as a framed knot. The original
idea is to form a link obtained by placing the knot inside a circle. This
was first shown to me by the sculptor Charles Perry. Later I saw this link
in several books on knots. This led me to the idea of a framed knot. From
a mathematical point of view the framed knot is not much different from
the original knot since it is easy to see that one can deform the framed
knot by lowering the added "frame" to obtain the original knot. However,
from a sculptural viewpoint, the framed knot is interesting since it has
a minimal surface that reverses form and space in the minimal surface of
the original knot. One can also suspend a wire knot in a wire circle using
an extra piece of wire to suspend the knot. This is the form that Charles
Perry introduced in order to obtain a sculpture where form and space are
switched. Perry forms the surface using flexible metal screen and then
applies automobile body putty on the screen. A mold is made and then the
minimal surface is cast in bronze. The idea of forming a sculpture from
the minimal surface of a framed knot or a knot in a circle is a very recent
development.
In Figure 7(b) we have a minimal surface
for a loop that consists of one Möbius band. It will now be shown that
there is a configuration of a trefoil knot that has a minimal surface consisting
of two Möbius bands that share part of their edges as in Figure
11(a) and also alternately cross over each other as in Figure
11(b). The knot is shown in Figure 12(a). In Figure
12(b) Möbius band 1 is shown and in Figure 12(c)
Möbius band 2 is shown. The intersection lines on the crossovers are indicated
by dotted lines.
If Figures 12 (b) and (c)
are combined, then we obtain the complete minimal surface shown in Figure
13. The two bands share parts of their edges and alternately cross
over each other. Band 1 crosses over band 2 at the top and band 2 crosses
over band 1 at the bottom. It is interesting to see each soapfilm Möbius
band twist as it crosses over the other Möbius band. The bands share edges
until they twist.
The knot in Figures 12 and 13 is the representation of a trefoil knot as the 2-3 torus knot. In general, given In general, the For example, the 3-4 torus knot has a minimal surface
consisting of three single twist Möbius bands. Suppose we color these three
bands red, white, and blue. Part of the surface will have the red band
and white band sharing an edge with the red band to the left of the white
band. The blue band will be crossing over the red and white bands. The
blue band will then twist and share an edge with the white band, where
the white band is now to the left of the blue band. The red band will now
twist and cross over the white and blue bands. This structure goes around
a central space so that each band twists once to cross the other two bands.
In order to picture this, we will first draw the 3-4 torus knot as in Figure
14. We begin with three points as in (a). We then
draw arcs as in (b). The knot is completed as in (c).
The corresponding minimal surface is shown in Figure 15.
Models of the multiple Möbius band minimal surfaces in Figures 13 and 15 are shown in the paper Knots and Multiple Möbius Bands published in Visual Mathematics. The behavior of minimal surfaces for configurations of
(
[1] M. Emmer, [2] [3] H. Levine, [4] L.D. Henderson, [5] N.A. Friedman, Hyperspace, Hyperseeing,
Hypersculptures, [6] N.A. Friedman, Hyperseeing, Hypersculptures,
and Space Curves, [7] N.A. Friedman, Hyperspace, Hyperseeing,
Hypersculptures (with figures), [8] N.A. Friedman, Geometric Sculptures
for K-12: Geos, Hyperseeing, Hypersculptures, |