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Higher-dimensional geometry considers the properties of figures with four or more dimensions. An object of zero dimensions is a single point. It needs no coordinates. The first dimension takes up the space of a line, and has only one coordinate (the x coordinate). The second dimension takes up the space of a plane, and has two coordinates, x and y. Examples of two-dimensional shapes are circles and squares. The third dimension is all space that we know of, and it uses three coordinates, x, y, and z. Examples of three-dimensional shapes are cubes, spheres, and cones. The fourth dimension is space with an extra dimension beyond the third, and it uses four coordinates, x, y, z, and w. Examples of four-dimensional shapes are the tesseract and the hypersphere.

Polytopes@

• Four-Space Visualization of 4D Objects Mathematical and computer programming oriented approach. Discusses wireframe rendering and ray-tracing.
• Fourth Dimension: Tetraspace A discussion of rotation, levitation, wheels, and bodies of water. Classification of "rotatopes" (cubes, spheres, and cylinders), and a Java applet to visualize projections and intersections of the shapes. Includes book reviews, a glossary, links, and a forum.
• Fun with Hyperspace The world of higher dimensions in math, physics, history and culture. Books, games and links.
• Hyper Dimensia Tutorial which aims to give readers an understanding of the fourth dimension by having them use a java applet to rotate a cube and a hypercube to see multiple views of the shapes at once.
• HyperCube by Harmen A short introduction to the fourth dimension, plus a hypercube java applet game.
• Hypercube's homepage Philosophy, math, and cosmology related to the fourth dimension.
• Hyperspace Travel Images and animations of a couple of four-dimensional objects, along with the math and source code used to produce them. Links to software to interactively visualize four-dimensional objects.
• Miscellaneous Mathematical Musings Introduction to dimensions, multi-dimensional geometric units, and formulas for calculating higher-dimensional phenomena.
• The 4D Web Page List of links to sites on people, books, art, java applets, games, and general information on the fourth dimension.
• The 4th Dimension Visual and mathematical descriptions of higher dimensional objects, with Java applets for displaying and rotating images such as Steiner surfaces, Klein bottles, and minimal surfaces.
• Thinking 4-d Focuses on helping the reader to conceive of the fourth spatial dimension, and in particular discusses hypercubes and hyperspheres.

 

 

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