Gallery Tour      Dr. Michael Brook

Introduction

The following is a summary of the mathematical principles used in the art works in this Gallery.     All work was done by art students enrolled in a course I taught at Community College of Philadelphia from 1983 to 1988.

1. Symmetry – Mirror Reflection

An original design is drawn. Lines are then drawn which act like the surfaces of mirrors, so that the “reflection” of the original design is also drawn on the opposite sides of these “mirror axes.”

The image generated by a kaleidoscope is an example of mirror symmetry (see image below – the black lines are mirror axes).

 

Examples of mirror reflection. Click on an image to see a full size image.

 

2. Circular Mirror Reflection

This distortion is not based on optical principles. It is derived from a straightforward extension of mirror symmetry to the case of a convex circular mirror axis. The mirror image P' of a point P is found by taking the distance from P to the circle, along a line PC connecting P to the center of the circle. Then this distance is repeated along the line in the interior of the circle. This establishes the location of P'. (This distortion is my own idea – M. Brook)

 

Examples of circular mirror reflection. Click on an image to see a full size image

 

3. Symmetry - Rotational

An original design is drawn and then copies of the design are rotated and placed in a circular pattern. 3, 4, 5 and 6 – fold rotational symmetries are represented here. The following chart shows how each type of symmetry is constructed:

Type of symmetry	First angle of  rotation	Copies of original design are placed at angles:
3 – fold	360°/3 = 120°	0°, 120°, 240°
4 – fold	360°/4 = 90°	0°, 90°, 180°, 270°
5 – fold	360°/5 = 72°	0°, 72°, 144°, 216°, 288°
6 – fold	360°/6 = 60°	0°, 60°, 120°, 180°, 240°, 300°

Examples of rotational symmetry. Click on image to see a full size view.

 

4. Perspective - Western

In the early 15^th Century, Italian artist Filippo Brunelleschi invented (some say re-invented) the principles of linear perspective. This system allows the artist to represent the three-dimensional world on a flat two-dimensional canvas. It uses a simple observation as its basis: objects appear smaller as they get farther away. This effect is called foreshortening. Since the real distance between two parallel lines remains constant, this distance appears to get smaller as the lines recede from an observer. If the visual images of the lines are extended, they will eventually intersect. The point of intersection is called the vanishing point and the horizontal line it lies on is called the horizon.

The intersection of parallel lines in perspective drawings contradicts the familiar notion that “parallel lines never meet.”  There is a field of mathematics known as non-Euclidean geometry in which parallels may intersect. Albert Einstein used non-Euclidean geometry in his theory of relativity.

Examples of Western perspective. Click on an image to see a full size view.

 

5. Perspective Anamorphosis

In perspective anamorphosis perspective is used for distortion, rather than for realism. The distortion is achieved in several ways: 1) perspective is applied to an object that is normally too small for perspective to affect, 2) the foreshortening is radical rather than gradual 3) the position of the horizon line is unusual – for example, it is vertical or oblique, 4) as in Faces1  Faces2 and Heads in the Gallery, several planes can be used.     

Examples of perspective anamorphosis. Click on an image to see a full size view.

6. Perspective - Isometric

This system of drawingn preserves true distances. For example, a real cube has edges of equal length. In an isometric drawing, the lengths of the edges remain equal. This is different from linear perspective, in which edges that recede from the viewer are drawn shorter to simulate the optical effect of things looking smaller in the distance. Because things do not get smaller in the distance in isometric perspective, parallel lines remain parallel. The preservation of true distances  is a desireable characteristic for engineering and technical drawing and isometric perspective is widely used for those purposes. Perceptually, howewever, when the receding edge of a cube is drawn with its true dimension it makes the top of a cube look ambiguously large. For this reason, isometric perspective is the basis of a class of optical illusions. In one such illusion, as the viewer stares at a figure, it seems to be alternately the outside and then the inside of a cube.

A related type of perspective is used in traditional Chinese and Japanese painting and computer game design.

Example of isometric perspective. Click on the image to see a full size view.

 

7. Cylinder Anamorphosis

In cylinder anamorphosis a drawing is made on a rectangular grid. Another grid is made where concentric circles correspond to the horizontals of the original grid and radii (spokes) correspond to the verticals. Then the artist transfers the drawing point by point from one grid to the other.

When the distorted drawing is complete, a cylindrical mirror is placed in the center. An undistorted image appears in the mirror.

This method of distortion was used during the Renaissance for entertainment, but also for politically risky caricatures and erotic art. The distortions were often extreme, making it necessary to know the secret of the mirror to decode them.

Examples of cylinder anamorphosis. Click on an image to see a full size view.

 

8. Curve Envelope

A curve envelope is a curve that is tangent to every member of a family of lines. For example, a family of lines can be generated as follows (see the diagram below): Draw the line from (0, 9) on the y-axis to (1, 0) on the x-axis. Then draw a line from (0, 8) on the y-axis to (2, 0) on the x-axis. Continue in this fashion so that all lines have the property that the y-intercept + x-intercept = 10.  The equations of these lines are:

.

The envelope of this family is a parabola (the red curve in the diagram below). Because the curve is tangent to every straight line, every straight line is also tangent to the curve. This accounts for the curved appearance of the family of lines, despite the fact that they are all straight.

Curve envelopes are the basis of string art or curve stitching, an educational practice invented by Mary Everest Boole (mathematics teacher, wife and student of mathematician George Boole) in 1904. She believed that mathematics should be learned with the body as well as the mind. In curve stitching, students make designs by sewing curve envelope families using colored thread.  

 

Examples of curve envelopes. Click on an image to see a full size view.

 

Curve Envelope Distortion

A curve envelope distortion is a distortion is based on a grid made from the family of lines described in (8). First a drawing is made on an ordinary rectangular grid and then the artist transfers the drawing point by point to the distorted grid (see figure below).

(This distortion is my own idea – M. Brook)

Example of a curve envelope distortion. Click on the image for a full size view.