Exploring Double Meanings in Geometric Figures

Jen-chung Chuan
Department of Mathematics
National Tsing Hua University
Hsinchu, Taiwan 300


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.


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:
1) let the vertices of a pentagon be given, to construct a new pentagon by joining the midpoints;
2) given a pentagon, to construct another one whose midpoints coincide with the corners of the given ones.
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:
1) if a hexagon is inscribed in a conic, then the three pairs of opposite sides meet in collinear points;
2) there is a conic passing through five given points; the illustration shows how to construct all others points of the curve.

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:
1) the figure consists of normal lines to the outer cardioid having the length the radius of curvature; consequently an envelope (its evolute) appears in the form of  the inner cardioid;
2) the region indicates where a string tautly unwound from the inner cardioid has swept, the  endpoint sweeping out the outer cardioid.

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:

1) the cardioid remains stationary while the pair of orthogonal tangents traveling around the curve;
2) the cardioid slides along two fixed orthogonal straight lines.
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:
1) it transforms the linear motion into a circular one;
2) it transforms the circular motion into a linear one.

Evelyn Sander has a webpage devoted to the discussion of  Peaucellier's cell.

Construction with Ruler Only

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:
1) given the midpoint of a line segment and given another point not on the segment, to construct, using the ruler alone, a straight line passing through the given point and parallel to the given line segment;
2) given a line parallel to a line segment, to construct, using the ruler alone, the midpoint of the line segment.
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:
1) the ellipse is shown to be rotate around rigidly while remain tangent to the deltoid internally;
2) the deltoid is to rotate around while touching the fixed ellipse all the time.

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
1) all circles seem to march in one direction only;
2) one half of the circles march towards left while the other half towards right.

Deltoid and Three-Cusped Epicycloid

This figure conveys two messages:
1) between the deltoid and the 3-cusped 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 3-cusped epicycloid.
Similar situations take place for the astroid and the 4-cusped epicycloid pair also:

Euclidean and Non-Euclidean Geometries

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 Non-Euclidean 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 Non-Euclidean geometry it shows the three common chords of pairs of circles meet are concurrent. This is the Non-Euclidean 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 three-dimensional counterpart:
1) the common external tangents to each pair of three different-sized circles meet in three collinear points;
2) the enveloping tangent cones of each pair of three different-sized spheres have collinear vertices.

You may consult Ogilvy's charming book "Excursions in Geometry" pp. 115-117 to see the complete explanation.

Here is another theorem belonging to the same category, known as Desargues' Two-Triangle 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' Two-Triangle Theorem given in Howard Eves' masterpiece "A Survey of Geometry".

Double Generation

Question: which of the following two statements is correct?
1) the deltoid is the locus of a point on the circumference of a circle which rolls round the inside of a fixed circle triple the radius;
2) the deltoid is the locus of a point on the circumference of a circle which rolls round the inside of a fixed circle 3/2 the radius.
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:
 "a proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate or capable of innumerable solutions." It is not a theorem, nor a problem, or rather it includes both. 

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:
1) If two circles admit a Steiner chain, they admit an infinite number, and any one of the direct tangent circles is a member of one chain;
2) under the Poincaré's  model of non-Euclidean geometry, if two circles are so related that a polygon can be inscribed to one and circumscribed to the other, then infinitely many polygons can be so drawn.
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 three-dimensional 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!


  1. H. S. M. Coxeter, Introduction to Geometry
  2. Heinrich Dorrie, 100 Great Problems of Elementary Mathematics
  3. Howard Eves, A Survey of Geometry
  4. Roger A. Johnson, Advanced Euclidean Geometry
  5. A.B. Kempe, How to draw a straight line; a lecture on linkage, reprinted by Chelsea in the collection "Squaring the Circle"
  6. Aleksandr Kostovskii, Geometric Constructions with Compasses Only
  7. E. H. Lockwood, A Book of Curves
  8. A. S. Smogorzhevskii, The Ruler in Geometrical Construction
  9. David Wells, Hidden Connections, Double Meanings
  10. Robert C. Yates, A Handbook on Curves and Their Properties
  11. Robert C. Yates, Geometrical Tools, a mathematical sketch and model book