The Visual Guide To Extra Dimensions: Visualizing The Fourth Dimension, Higher-Dimensional Polytopes, And Curved Hypersurfaces (A Fourth Dimension of Space)

a tesseract

• Find numerous illustrations of or relating to a fourth dimension of space.
• This guide to extra dimensions is accessible while still being clear and detailed, and is focused on geometry.
• The author, Chris McMullen, has a Ph.D. in particle physics, has over 20 years of university teaching experience, is an author of the Improve Your Math Fluency series, and is a coauthor of a paper on the collider physics of extra dimensions with over 100 citations.

Ref.: C. Macesanu, C.D. McMullen, and S. Nandi, "Collider Implications of Universal Extra Dimensions," Phys. Rev. D66, 015009 (2002).

a 2D being "passing" a telephone pole

• The book begins with a fascinating tour of the lower dimensions.
• Many simple activities that we take for granted in 3D space would be impossible or drastically different in a 2D world, like hammering a nail through a board or digging a tunnel.
• By trying to imagine what it would be like to be a 2D being in a 2D world trying to contemplate a 3D world, we can gain insight into a possible fourth dimension of space.

six orthogonal planes intersecting in 4D space

• In ordinary 3D space, three planes (xy, yz, and zx) are mutually perpendicular when they intersect.
• In 4D space, six planes (xy, yz, zx, wx, yw, and zw) are mutually perpendicular when they intersect.
• In the diagrams above, the two different diagonal directions represents the w and z axes, which are perpendicular to one another and are also perpendicular to x and y (which are horizontal and vertical).

a 4D staircase (hyperstairs)

• The idea of perspective, which can be used to create optical illusions when drawing 3D objects on 2D paper, becomes even more difficult to interpret when it is used to draw 4D objects on 2D paper.
• As a result, the 4D stairs shown above are more difficult to understand than a drawing of 3D stairs. However, there are ways to help interpret diagrams such as this and to gain insight into a possible fourth dimension of space.