What Should a Four Dimensional “Cube” Look Like? — -the Isometric “Tesseract”

Coining the word “tesseract” is generally credited to Charles Howard Hinton, a 19th century scientist and science fiction writer who had a particular interest in the concept of a fourth dimension. In 1880, he wrote an article called “What is the Fourth Dimension?” in which he suggests that the fourth-dimension a) exists and b) could be time-related. He then introduced the idea of visualizing this fourth dimension with cubes. He tried out different words to describe it and came up with “tesseract” for his 1888 book, A New Era of Thought. His 1906 book, The Fourth Dimension describes his concept in excruciating detail. Unfortunately, I don’t think that it is a satisfying solution to the problem of visualizing something in three dimensions that is to the cube what the cube is to the square in two dimensions. In any case, his idea of the 4-dimensional cube was something quite different from the box-within-a-box “tesseract” that is used these days to illustrate the concept.

Do we ordinarily draw a cube, which we might describe as a 3-dimensional square, as a square within a square?

Unless we are depicting it in one sort of perspective view, we draw a 2-dimensional representation of a cube in a manner that will preserve some of the characteristics that make it a cube, such as edges of equal length, sets of parallel edges and parallel faces, equal angles between edges, and edges perpendicular to each other. We can’t get all these things in one drawing; dropping from three dimensions to two always causes a loss of some amount of information. There are, however, depictions three dimensions in two dimensions that preserve enough information that they are useful.

One of these is the isometric drawing, which, as its name implies, preserves dimensions in the three principal directions. The angle between the coordinates is set at 120 degrees, so that they are equal, although not perpendicular.

A 2-dimensional isometric construction of a cube with opaque faces is easily seen as a cube, even though it is strictly simplified. It gets a little more difficult to visualize if the faces are transparent; the geometry of a hexagon with three diagonals takes over. Part of the visual confusion results from the fact that two of the vertices of the cube lie exactly at the same point in the construction.

We can construct a 3-dimensional image of a 4-dimensional shape in a similar fashion that preserves some of the relationships between its parts. Of course, we must expect that the 3-dimensional model will lose information, just like what happens in the representation of the cube in a 2-dimensional drawing.

To create the four cardinal axes of the 4-dimensional model I use a coordinate system that is based on the diagonals of a cube. The origin is at the center of the cube, where the four diagonals intersect, and the coordinates radiate at equal angles from each other.

We can build the model of the 4-dimensional shape in the same way that we might build an isometric cube. Starting with a point at the origin of a 4-dimensional x, y, z, w coordinate system, we can construct a line by moving the point a certain distance along the x and y axes and add lines parallel to them to form a square. We can offset the square along the z axis and connect it to the first square to form a cube. In our model the edges of this cube are not perpendicular to each other. They appear to be rhombuses, and the relationship to our rhomboid “cube” to a cube is much the same as the relationship of a rhombus-shaped “square” drawn in isometric is to a square.

Just as we extended the square along the z axis to form a cube, we can now extend the cube along the w axis to generate a hypercube with edges that are parallel to each other and all of equal length. The exterior form of the model is a rhombic dodecahedron.

As in the isometric drawing of a cube, the 3-dimensional construction of the hypercube has two vertices that appear to be in exactly the same location, at the center of the model. It is apparent, however, that the model has more shared vertices than this. Just as the square has four linear sides and the cube six square faces, the hypercube has eight cubical “faces”. Each “face” of the rhombic hypercube can be seen to be attached to six other cubical “faces”, just as would be expected. Just as the three faces on the back of the cube drawn on an isometric plane are seen to have the same area as the three front faces that cover them, four “front” cubical faces of the hypercube coincide in volume with the remaining “back” cubical faces.

Here is an exploded view of the rhombic hypercube that shows how the cubes that make it up are arranged.

As in the isometric drawing of a cube, the 3-dimensional construction of the hypercube has two vertices that appear to be in exactly the same location, at the center of the model. It is apparent, however, that the model has more shared vertices than this. Just as the square has four linear sides and the cube six square faces, the hypercube has eight cubical “faces”. Each “face” of the rhombic hypercube can be seen to be attached to six other cubical “faces”, just as would be expected. Just as the three faces on the back of the cube drawn on an isometric plane are seen to have the same area as the three front faces that cover them, four “front” cubical faces of the hypercube coincide in volume with the remaining “back” cubical faces.

Here is an exploded view of the rhombic hypercube that shows how the cubes that make it up are arranged.

Just as the 2-dimensional isometric representation of the cube shows three faces, or half the area of the cube it represents, the rhombic hypercube’s 3-dimensional representation has half the volume of the 4-dimensional object that it represents.

Here is a2-dimensional isometric view that shows all the edges in each of the four directions. All the edges are of equal length and all the faces, wherever the are on the model, are congruent.

It could be built with matchsticks as a 3-dimensional model.

I think that this is a more useful and accurate representation of a 4-dimensional “cube” than the first picture in this article.

Squares can tessellate a 2-D plane. Cubes can tessellate a 3-D space. Rhombic dodecahedrons can also tessellate a 3-D space. It is not hard to imagine that 4-D cubes should tessellate a 4-D space.

It is possible to form other coordinate arrangements that have more than three or four coordinates. To make them as regular as possible, the starting point should be a regular figure, as it is with the 4-D system described above, which is based on the diagonals of a cube. For instance, lines that connect the midpoints of opposite edges of a cube, or the centers of the faces of a regular dodecahedron, form an equally arranged six coordinate system. Perhaps we can develop a hyperhypercube model.…