Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

# Glossary

## Dimension

Dimension is how mathematicians express the idea of degrees of freedom—aspects of an object that can be measured separately.

## Flat Land

Points in two-dimensional space require two numbers to specify them completely. The Cartesian plane is a good way to envision two-dimensional space.

## Hypercube

The hypercube is the four-dimensional analog of the cube, square, and line segment. A hypercube is formed by taking a 3-D cube, pushing a copy of it into the fourth dimension, and connecting it with cubes. Envisioning this object in lower dimensions requires that we distort certain aspects.

## Hyperland

A point in four-space, also known as 4-D space, requires four numbers to fix its position. Four-space has a fourth independent direction, described by "ana" and "kata."

In Euclidean four-space, our standard notions of Pythagorean distance and angle via the inner product extend quite nicely from three-space.

## Hypersphere

A sphere can be thought of as a stack of circular discs of increasing, then decreasing, radii. The process of slicing is one way to visualize higher-dimensional objects via level curves and surfaces. A hypersphere can be thought of as a "stack" of spheres of increasing, then decreasing, radii.

## Line Land

A point in one dimension requires only one number to define it. The number line is a good example of a one-dimensional space.