The Fourth Dimension

Explorations

Begin learning about flatland and the fourth dimesion with:

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.

A point (0-dim), a segment (1-dim), a square (2 dim) and a cube (3-dim)

In the image above we see a point, a line segment, a square and a cube. These shapes are 0-, 1-, 2-, and 3-dimansional respectively. One could ask: what is the next shape in that sequence? In other words, is there something like a 4-cube.

To construct a model of this 4-cube, also known as a tesseract, we can follow the procedure of creating higher dimensional shapes. Think about extending the shape we have into a dimension perpendicular to those we already have. For instance, starting with a line segment we can draw an exact copy of the segment in the plane and then connect the corresponding vertices. This creates a square.

Using two copies of a segment (1-dim) we construct a square (2-dim) by connecting the corresponding vertices

Similarly, connecting corresponding vertices on two copies of the square will result in a cube.

Given two cubes we can connect the corresponding vertices and construct a 4-dimensional cube or tesseract.

A 4-dimensional cube. The figure on the right shows the two cubes (yellow and blue) whose corresponding vertices have been connected.

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:

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 three different ways, while the square and the pentagon result in two more platonic solids:

• Four triangles will form a tetrahedron.
• Eight triangles will form a octahedron.
• Twenty triangles will form an icosahedron.
• Six squares will form a cube.
• Twelve regular pentagons will form a dodecahedron.

In "Stars" we see two chameleons inside a shape made up of three octahedra. In the background - floating around in space - are some platonic solids and several other geometric 3-dimensional figures. Wikipedia has a short article with links describing some of the more exotic geometric shapes that are shown in the background. [Stars (M. C. Escher)]

Flatland

Flatland was written in the 19th century, and is both a satire on Victorian Society and an exploration of the mathematical notion of dimensions. We read this story to develop some ideas about how to think about the 4th dimension. Even though we do live in the 4-dimensional space-time, most people are not comfortable with the 4th dimension at first. There are two questions we are interested in. What would a 4-dimensional being look like if it interacted with us? What would our 3-dimensional world look like if someone moved us into the 4th dimension? One way to think through these questions is to first ask them with all the dimensions dropped down a bit. What would a 3-dimensional being look like if it interacted with 2-dimensional beings (i.e. Flatlanders)? Or, what would a 2-dimensional being look like if it interacted with a 1-dimensional being? What would the 2-dimensional world look like if someone moved a Flatlander into the 3rd dimension? Abbott answers all of these questions in the book Flatland.

The 3-dimensional sphere appeared one 2-dimensional slice at a time. The sphere would first appear as a dot, and then grow into ever increasingly large circles. After reaching its biggest circumference, the circles would shrink back down to a point again. But the important part here is that the flatlanders could only see a 2-dimensional cross-section. Their 2-dimensional eyes and brains were not used to thinking about or seeing 3-dimensional beings.

Similarly, we would expect to see 3-dimensional cross-sections of any 4-dimensional beings.

When A. Square (the main character in Flatland) traveled to Lineland, he could see all of their world at once. The King of Lineland at first doesn’t know who is talking to him, because he can’t see Mr. A. Square at all. Later in the story A. Square moves into Spaceland, and is able to look down upon his own world.

A. Square is able to see the interior and exterior of his house at the same time. He can also see his family moving about the house. If you look carefully at the picture, you would see that A. Square can also see inside his relatives. Similarly, if we were moved into the 4th dimension, we might be able to see our world all at once. We would see the interior and exterior of our houses simultaneously, and we would also be able to see all around people.