11B: Perspective and Symmetry

Two of the three aspects of Art that relate directly to Mathematics!

Perspective

• Use of geometrical ideas developed by the Greeks.– Knowledge lost, then regained during

Renaissance!

What do you see?

How do you paint it?

Perspective

• Brunelleschi (1377-1466)

• Alberti (1404-1472) ↑

Two painters credited with first developing a method of painting perspective that involved geometrical thinking.

Perspective

• Geometrical Method of Drawing Perspective Involves…– 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.

Perspective

– Horizon Line: A horizontal line passing through the Principal Vanishing Point. The Line at which parallel lines meet.

History!• Leonardo da Vinci (1452-1519):

Contributed to the science of perspective through his paintings.

• Albrecht Durer (1471-1528): Developed woodcutting techniques for showing perspective

• Jan Vredman de Vries (1527-1604): Made extensive sketches to study perspective

Symmetry

• Refers to: Balance in a picture or shape, or repetition of patterns.

• IN MATHEMATICS: Symmetry is a property of objects that remain unchanged after certain operations. (Rotation, Reflection, and Translation.)

Symmetry

• Reflection Symmetry: An object that remains unchanged when reflected across a straight line.

• Rotation Symmetry: An object that remains unchanged when rotated through some angle about a point.

Symmetry

• Translation Symmetry: An object that remains that same when repeated, creating a pattern.

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Symmetry

• Symmetry in Art:

Symmetry• Tilings: covering a

flat area with

geometrical shapes;

An arrangement

of polygons into

a pattern.

Perspective And Symmetry

• Homework!– Questions: 1, 24, 30, 32, 37