# Depth and Perspective

## Introduction to Dimensions

Webster’s Dictionary gives a description of dimensions 1, 2 , 3, and 4. We will also give a description of the 0th dimension. Space of zero dimensions: A space that has no length breadth or thickness. An example is a point. Space of one dimension: A space that has length but no breadth or thickness; a straight or curved line. Space of two dimensions: A space which has length and breadth, but no thickness; a plane or curved surface. Space of three dimensions: A space which has length, breadth, and thickness; a solid. Space of four dimensions: A kind of extension, which is assumed to have length, breadth, thickness, and also a fourth dimension.

Space of five or six, or more dimensions is also sometimes assumed in mathematics.

You have seen examples of these objects al your life, but are probably not used to thinking about them in this way. A pin-prick can be thought of as 0-dimensional. To the naked eye it has no length, width or height.

Whenever you draw a line on a piece of paper, you are drawing a representation of a 1-dimensional object. We only measure one direction. We will ask for the length of a line-segment, but we would not ask questions about its width or height. We think of a line as simply not having any width or height.

Whenever you measure the area of a geometric object you are measuring something 2-dimensional. Think about how you would find the area of a rectangle. You would measure the length and the width and multiply the two, right? You measure the two dimensions, and so we think of the rectangle as 2-dimensional.

Anything 3-dimensional will require 3 measurements. Hence the volume of a box is considered 3-dimensional. It has length, width, and height.

Now consider making an appointment with someone at a 10 story building. You will have to tell them where to meet and when. The location is 3-dimensional, because we need 3 coordinates to find a place in space: longitude, latitude and height. But this means that to determine our place and time of meeting we require 4 pieces of information: longitude, latitude, height AND time. This means that our appointment is something 4-dimensional. This idea of space and time, appropriately named 4-dimensional space-time, was used by Einstein when he brought forth his theory of relativity.

## Escher and Dimension

Escher looked at the interplay between 3-dimensional objects and their 2-dimensional depictions. He used the play on dimensions to create several interesting prints. Some examples include:

- Cycle
- Day and Night
- Drawing Hands
- Liberation
- Metamorphosis I
- Metamorphosis II
- Metamorphosis III
- Reptiles

A famous print by Escher showing the contrast between 2 and 3 dimensions is the print named drawing hands. The hands in the print are clearly 3-dimensional. The hands and the pencils are shown as existing in space. It gets more interesting when we move our eye to the wrists and the lower arm. Here Escher transitions to a 2-dimensional image. The underlying piece of paper is depicted as entirely flat.

Escher's tessellations are all 2-dimensional. He referred to them as "Regular Divisions of the Plane" (Regelmatige Vlakverdeelingen in Dutch) and they all depict patterns that decorate a nice flat, 2-dimensional surface.

Escher also studied regular 3-dimensional shapes. Some examples include:

First, there are the platonic solids. If we experiment with regular polygons and try to build 3-dimensional shapes, then there are only 3 regular polygons that can be used by themselves.

The triangle can be used in different ways: four triangles will form a tetrahedron, eight triangles will form a octahedron, and twenty triangles will form an icosahedron.

If we use a square, then clearly six of them will form a cube.

Finally, twelve regular pentagons will form a dodecahedron.

In "Stars" we see the platonic solids and several other geometric 3-dimensional figures.