Image 1 Caption and Image 2 Caption: Animations showing from 2 different directions a tesseract being rotated in 4-dimensional Euclidean space projected into 3-dimensional Euclidean space.

Features:

1. The tesseract (AKA 4-cube) is a 4-dimensional hypercube (AKA n-cube) which is the n-dimensional analog of a cube (AKA 3-cube).

   Note the following well known hypercubes with special names: point (0-cube), line segment (1-cube), square (2-cube), and, of course, cube (3-cube) (see Wikipedia: Hypercube: Elements).

2. Now we humans have difficulty perceiving 4-dimensional shapes, but there are ways of approaching perception.

   The projection of a 4-dimensional shape into a 3-dimensional Euclidean space (i.e., 3-dimensional hyperplane) is a 3-dimensional shape.

   Humans can understand 3-dimensional shapes thanks to stereopsis, experience, imagination, and lots of other things (see Wikipedia: Stereoscopy: Background).

   It's all pretty tricky to analyze, but we mostly do it pretty easily. We can be fooled by optical illusions, of course.

   

3. So as the animations illustrate, we can perceive the projections into 3-dimensional Euclidean space of the tesseract.

   If the tesseract were NOT rotating, it would just be a somewhat unusual 3-dimensional shape.

   The rotation in 4-dimensional Euclidean space makes the projected shape morph.

   See Wikipedia: Tesseract for a fuller description of the animations.

4. The ordinary analog to the animations is the just rotating a cube. NOT considering stereopsis, all we see is a 2-dimensional distorted squares morphing, disappearing, and reappearing. It would all be very mysterious if we did NOT have stereopsis, experience, imagination, and lots of other things (see Wikipedia: Stereoscopy: Background).

5. See also 4th Dimension - Tesseract, 4th Dimension Made Easy - Carl Sagan | 9:29: Carl Sagan discusses Flatland (1884) (written by Edwin Abbott Abbott (1838--1926)) and the tesseract (AKA 4-cube) (i.e., one kind of hypercube). Too long for the classroom.

6. By the by, Robert A. Heinlein (1907--1988) wrote a scifi story, "---And He Built a Crooked House---" about an architect who builds a tesseract house. It's a fantasy about unusual geometry and NOT about a realistic tesseract house.

Images

1. Credit/Permission: User:JasonHise, 2007 / Public domain.
   Image link: Wikimedia Commons: File:8-cell.gif.
   
2. Credit/Permission: User:JasonHise, 2007 / CC BY-SA 1.0.
   Image link: Wikimedia Commons: File:8-cell-orig.gif.