The Tesseract: Folding and Unfolding a Simple 4D Hypercube

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A tesseract is the equivalent of a cube in four dimensions; in other words, it is a four-dimensional hypercube. As G. Olshevsky comments in Glossary for Hyperspace, a hypercube fits into “n-space” for any value of “n”. Human laziness leads few people to think in five dimensions or beyond. Therefore a hypercube is often considered to refer to merely four dimensions.

To Develop a Tesseract

A good place to start developing a tesseract is with this image. This paragraph will follow the series of five drawings, going from top to bottom.

"Develop a Tesseract" by Mike DeHaan

“Develop a Tesseract” by Mike DeHaan

Start with a single point, which has no “size”. A point is not a tiny circle, because a circle has the length of its circumference and the area it covers.

If one could hammer down the point at its starting location, and then drag it along in one direction, it would leave a trail called a “line”. That is, if it were large enough to be nailed down, and still have some surface left over to grab. It really does not have those properties, so it is easier to just draw a line with a pencil.

Still, a line is the extension of a point in one dimension. In the diagram above, it extends in an east-west direction. Now the line has a length.

If one drags a line at right angles to its length, the new shape is a rectangle. Dragging it exactly the same distance north-south as its east-west length creates a square.

Drag the square into the up-down direction, “coming off the computer screen” (or the page) for the same distance as the length of its original line. Now it has become a three-dimensional cube.

Finally, drag the square into the fourth dimension, at right angles to the original three dimensions, for the same distance as the length of the original line. Now it has become a four-dimensional tesseract. This is the final figure in the above image. The point of view shows the inside of the 3-dimensional cube nearest to the observer; the grey shadows show three of the planes of that cube. Of course, the diagram shows an unfolded tesseract.

Folded or Unfolded Shapes

The image below shows an eight-cell tesseract, except that the lines at the edges are rounded pipes rather than one-dimensional lines. This is also an unfolded tesseract.

"Eight Cell Tesseract Image" by mmhydrazinen04

“Eight Cell Tesseract Image” by mmhydrazinen04

The perspective for this image shows that the nearest cube looks more like a section from a pyramid. The base, nearest the observer, appears larger than the “head” attached to the central cube, due to the perspective of the observer.

Really, this is the same type of illusion that is accepted in the drawing of the cube in the first diagram. The top of the cube, or the “lid on the box”, is drawn as a parallelogram rather than as a square. It is just a trick of perspective.

J. Köller puts this discussion into perspective in “Hypercube, Tesseract” with some very nice diagrams.

"Unfolded Cube" by Mike DeHaan

“Unfolded Cube” by Mike DeHaan

 

This image depicts the process of unfolding a cube. The “front” surface, marked with the green blob, moved to become the top square. The blue blob shows the “lid of the box” unfolded to become the second square from the top.

The brown and orange blobs show the “sides of the box” unfolded to become the left and right squares. Finally, the “floor of the box” is now the bottom square, with the purple blob.

 

 

Building Tesseracts for Fun and Profit

"Messy Toothpick Tesseract" by Mike DeHaan

“Messy Toothpick Tesseract” by Mike DeHaan

The “Messy Toothpick Tesseract” demonstrates that it is possible to build an unfolded four-dimensional rectangular object in ordinary three-dimensional space. Hot glue worked better than “sticky tack” to hold the toothpicks together. The strands of glue give the structure an illusion of cobwebs and timeless neglect.

This image may demonstrate why Robert Heinlein’s story about a four-dimensional home was called “And He Built a Crooked House”.

Enjoy this video representation of a transparent tesseract:

References

George Olshevsky, “Glossary for Hyperspace”, referenced May 13, 2011.

Jürgen Köller, Mathematische Basteleien, “Hypercube, Tesseract“, copyright 2001, referenced May 13, 2011.

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