Jenchung Chuan
Department of Mathematics
National Tsing Hua University
Hsinchu, Taiwan 300
jcchuan@math.nthu.edu.tw
Abstract
Mathematics is full of riddles. As a result, pictures describing mathematics
often contain double meanings. Mathematics is kept alive and stimulating
in part due to the multiplicity of ideas represented by the geometric images.
In what follows we are to reveal, through the dynamic geometry environment
set up by CabriJava, double
meanings hidden in a wide range of static geometric figures. 
Introduction
Why is a geometric figure important? A geometric figure clarifies a theorem,
motivates a proof, stimulates the thinking process, sums up a lengthy animation,
provides a counterexample to a wild conjecture, or just plainly announces
the existence of a significant piece of mathematics.
Why is a geometric figure interesting? It is interesting often because
it carries double meanings.
"Given" and "To Construct" Switched
A geometric construction problem has three parts: "Given", "To Construct",
and the construction itself. Imagine that we come across some ancient script
on geometry in which the written language is indecipherable but the illustration
remains intact as shown:
By switching the "Given" and the "To Construct" parts, we see that the
picture may have these two interpretations:
This tiny example shows that dynamic geometry is at least twice as interesting
as the traditional one.
In the same vein, consider the figure illustrating the Pascal theorem:
The picture carries two messages:
Evolute and Involute
Involute is the path of a point of a string tautly unwound from the curve.
Evolute of a curve is the locus of its center of curvature. Here is a figure
consisting of line segments each joining a point of a cardioid with its
center of curvature:
The figure may be interpreted in two ways:
The same phenomenon exists in other epicycloids and hypocycloids as
well:
Since the epicycloid and its involute can be transformed into one another
by a central similarity, an interesting nested pattern may be constructed
after this method:
Cardioid and Perpendicular Tangents
Here is a static figure showing two perpendicular tangent lines of the
cardioid:
There are two ways to regard the figure as a particular instance of
a sequence of shots:
The similar phenomenon exists in other epicycloids and hypocycloids as
well:
Peaucellier Cell
The design of the linkage known as the Peaucellier cell was the first
mechanical inversor ever awarded:
The device may be used in two ways:
Evelyn Sander has a webpage
devoted to the discussion of Peaucellier's cell.
Stimulated by the theory of perspective, the study of the constructive
power of a ruler was carried out in full during the 19th century. Geometric
figures so constructed usually consist of straight lines only. Such figures
can be borrowed to create amusing puzzles by asking: what was the process
of construction? Thus, for example, if we regard this figure as the solution,
what then, is the question? There are two
possibilities:
Here is a book devoted to constructions of this sort:
A. S. Smogorzhevskii, The
Ruler in Geometrical Construction.
For those who are curious, there is another book translated from Russian
on the subject of geometric constructions with the compasses alone:
Aleksandr Kostovskii, Geometric
Constructions with Compasses Only
Ellipse and Deltoid
This picture does not appear impressive until turned into a dynamic one:
Depending how the "camera" is manipulated, we may build these
two
sequences of animation:
Coaxal Systems
Can you make circles in this illustration of the coaxal system move?
Based on two different
principles of design, the static picture can be turned into a dynamic
one with
Deltoid and ThreeCusped Epicycloid
This figure conveys two messages:
1) between the deltoid and the 3cusped epicycloid there are circles
having center on the base circle and tangent to both; 
2) there is a family of circles enveloping both the deltoid and the
3cusped epicycloid. 
Similar situations take place for the astroid and the 4cusped epicycloid
pair also:
Consider this interesting figure:
The figure can be regarded as either an illustration of a theorem in Euclidean
geometry, or an illustration of a theorem in NonEuclidean geometry.
1) as a figure in the Euclidean geometry, it shows the three arcs each
orthogonal to the big circle and passing through the points of intersection
of two circles, meet at one point; 
2) as a figure under the Poincare model of the NonEuclidean geometry
it shows the three common chords of pairs of circles meet are concurrent.
This is the NonEuclidean version of this illustration in Euclidean geometry: 
2D Phenomena Explained Through 3D
There are interesting theorems who proofs can be given when the 2D figures
so drawn be viewed as 3D figures. One such famous results is Monge Theorem:
Here are the statements of the original theorem and its threedimensional
counterpart:
1) the common external tangents to each pair of three differentsized
circles meet in three collinear points; 
2) the enveloping tangent cones of each pair of three differentsized
spheres have collinear vertices. 
You may consult Ogilvy's charming book "Excursions
in Geometry" pp. 115117 to see the complete explanation.
Here is another theorem belonging to the same category, known as Desargues'
TwoTriangle Theorem:
Depending on the 2D or the 3D point of view, the figure says:
1) copolar triangles are coaxial, and conversely; 
2) copolar triangles in space are coaxial, and conversely. 
This is just one of the four different proofs of Desargues' TwoTriangle
Theorem given in Howard Eves' masterpiece "A
Survey of Geometry".
Double Generation
Question: which of the following two statements is correct?
Answer: Both are correct!
With the traditional printing technology this is all that can be illustrated:
Under the dynamic geometry environment it is highly stimulating to construct
the phenomenon known as the "double
generation" which states that every cycloidal curve may be generated
in two ways: by two rolling circles the sum, or difference, of whose radii
is the radius of the fixed circle.
Steiner Porism, Poncelet Porism
According to the Webster's 1828 dictionary, a porism is defined this way:
So much for an attempt to define a respectable mathematical result!
One thing is clear: any statement qualified to be named a "porism" must
have double meanings.
This illustration of the Steiner porism
carries double meanings as follows:
Statement 1) is known as the Steiner's porism, while statement 2) is known
as the Poncelet's porism.
Front and Back
When the graph of the plane graph of the function y = sin 2x were wrapped
around a cylinder, it appears as:
The reason a primitive drawing
such as this appears as a threedimensional object is because in our mind
we have assigned the notion "front" and "back" to the crucial portions
of the figure. But then, there are two such possibilities!
References

H. S. M. Coxeter, Introduction
to Geometry

Heinrich Dorrie, 100
Great Problems of Elementary Mathematics

Howard Eves, A
Survey of Geometry

Roger A. Johnson, Advanced
Euclidean Geometry

A.B. Kempe, How to draw a straight line; a lecture on linkage, reprinted
by Chelsea in the collection "Squaring
the Circle"

Aleksandr Kostovskii, Geometric
Constructions with Compasses Only

E. H. Lockwood, A
Book of Curves

A. S. Smogorzhevskii, The
Ruler in Geometrical Construction

David Wells, Hidden
Connections, Double Meanings

Robert C. Yates, A
Handbook on Curves and Their Properties

Robert C. Yates, Geometrical
Tools, a mathematical sketch and model book