1. Mathematics and the Arts11

2. Perspective and Symmetry
Unit 11B
Perspective and Symmetry

3. Connection Between Visual Arts and Mathematics
At least three aspects of the visual arts relate directly to mathematics:
Perspective
Symmetry
Proportion

4. Side view of a hallway, showing perspectives.

5. Perspective and Vanishing Point
An artist’s view of the hallway, showing perspectives. The lines L1, L2, L3, and L4, which are parallel in the actual scene, are not parallel in the painting.

6. Perspective
Lines that are parallel in the actual scene, but not parallel in the painting, meet at a single point, P, called the principle vanishing point.
All lines that are parallel in the real scene and perpendicular to the canvas must intersect at the principal vanishing point of the painting.
Lines that are parallel in the actual scene but not perpendicular to the canvas intersect at their own vanishing point, called the horizon line.

7. Example

8. Example

9. Symmetry
Symmetry refers to a kind of balance, or a repetition of patterns.
In mathematics, symmetry is a property of an object that remains unchanged under certain operations.

10. Symmetry
Reflection symmetry: An object remains unchanged when reflected across a straight line.
Rotation symmetry: An object remains unchanged when rotated through some angle about a point.
Translation symmetry: A pattern remains the same when shifted to the left or to the right.

11. Example Identify the types of symmetry in the star. Solution
The five-pointed star has five lines about which it can be flipped (reflected) without changing its appearance, so it has five reflection symmetries. Because it has five vertices that all look the same, it can be rotated by 1/5 of a full circle, or 360°/5 = 72°, and it still looks the same. Similarly, its appearance remains unchanged if it is rotated by 2 × 72° = 144°, 3 × 72° = 216°, or 4 × 72° = 288°.
The star has four rotation symmetries.

12. Tilings (Tessellations)
A tiling is an arrangement of polygons that interlock perfectly without overlapping.
Regular Polygon Tessellations

13. Tilings (Tessellations)
Some tilings use irregular polygons.
Tilings that are periodic have a pattern that is repeated throughout the tiling.
Tilings that are aperiodic do not have a pattern that is repeated throughout the entire tiling.

14. Example
Create a tiling by translating the quadrilateral shown in Figure As you translate the quadrilateral, make sure that the gaps left behind have the same quadrilateral shape. Solution We can find the solution by trial and error, translating the quadrilateral in different directions until we have correctly shaped gaps. Figure shows the solution. (next slide)

15. Example
Note that the translations are along the directions of the two diagonals of the quadrilateral. The gaps between the translated quadrilaterals are themselves quadrilaterals that interlock perfectly to complete the tiling.