Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Dimension is how mathematicians express the idea of degrees of freedom—aspects of an object that can be measured separately.
Points in two-dimensional space require two numbers to specify them completely. The Cartesian plane is a good way to envision two-dimensional space.
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.
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.
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.
A point in one dimension requires only one number to define it. The number line is a good example of a one-dimensional space.
A point in three-dimensional space requires three numbers to fix its location.