Mathematical reference
- Vertices in n dimensions
- 2^n
- Edges in n dimensions
- n × 2^(n−1), for n ≥ 1
- Tesseract
- 16 vertices, 32 edges, 24 square faces, 8 cubical cells
Lesson transcript
This is the text of the narrated lesson. References to turning, highlighting or changing an object describe the linked interactive lesson; its demonstrations are not embedded in this reading page.
Keep every vertex accounted for
Begin with the four vertices of a square. As they spread into an array, each marker keeps its identity. We are rearranging the display so the parts are easier to count. We are not stretching the square into a new mathematical shape. Four corner positions belong to the original square.
Here are the eight vertices of a cube. Recall how two squares made that cube: four vertices came from one square and four from its copy. There are two choices in each of three coordinates, giving two times two times two positions. The array helps us check that we have included every corner.
The tesseract has sixteen vertices. Its fourth coordinate doubles the cube’s eight choices, just as copying the cube doubled its vertices. The displayed markers belong to one indexed tesseract. The labels identify those original vertices; they are not additional coordinate values, and changing the display arrangement does not create new vertices.
One dimension further gives thirty-two vertices. Each new coordinate adds another choice between two endpoints, so the vertex count doubles at every step. For an n-dimensional cube, this is two to the power n. That expression summarizes the repeated doubling we can already see in the square, cube and tesseract.
Choose a direction for an edge
Now look at a cube’s edges, separated from their original arrangement. There are twelve in all. An edge varies along one coordinate while the other two coordinates stay fixed. We will group edges by their varying direction, which gives a more organized count than tracing every line in a crowded projection.
Keep just one direction family. Four edges remain. Along each edge, one coordinate changes from one endpoint to the other. For each of the other two coordinates there are two fixed choices. Two times two gives four parallel edges in this family, even though perspective can make their projected directions look different.
Here is a different direction family of the cube. It also contains four edges. There are three possible varying coordinates, so there are three families with four edges each: twelve altogether. We count an edge once by choosing its direction and its fixed coordinates, rather than once from each of its endpoints.
In a tesseract, one edge direction has eight members. There are three remaining coordinates to fix, with two choices each. Four possible directions give four families of eight edges, or thirty-two edges in total. This view shows one of those four families, so the eight visible edges are only part of the whole inventory.
A square needs two free directions
Turn to one square face of a cube. Two coordinates vary across this face, while the third stays fixed. This is what makes the face two-dimensional. Moving our camera does not change those coordinate roles. We can choose its orientation by choosing which two directions are free to vary.
Here are both faces in that orientation family. The remaining coordinate can be fixed at either endpoint, so the family contains two opposite squares. They belong to the same cube, even after the display separates them. A pair of free directions identifies the orientation; a fixed endpoint identifies which member we have selected.
Now show all six cube faces. There are three ways to choose two free directions from the cube’s three directions, with two faces in each orientation. Three times two gives six. Their separate square boundaries repeat the edges where faces meet, so do not count this display as twenty-four distinct cube edges.
For the tesseract, one square orientation has four members. Choose two free directions, then fix the other two coordinates in one of four combinations. There are six ways to choose the free pair from four directions. Six families with four squares each give twenty-four square faces in the complete tesseract.
Cubes within the tesseract boundary
Come close to one cubic cell of the tesseract. Three coordinates vary within this cell, while the fourth remains fixed. Its edges and square faces are the familiar parts of a cube. The shape we see here is projected, but the cell in the original four-dimensional construction is a cube.
Show the two cells in this orientation family. They use the same three free directions, with the remaining coordinate fixed at opposite endpoints. These are opposite cubic cells. As with the opposite square faces of a cube, the pair comes from the two possible positions along the one coordinate that stays fixed.
Separate all eight cubic cells for counting. There are four choices for the coordinate that stays fixed, and two endpoint choices for each. Four times two gives eight cells. They make the tesseract’s boundary. The array opens their shared faces for inspection; it does not add eight independent interiors to the original object.
Move into one cell in that array and trace its six square faces. Eight cells contribute forty-eight face appearances, but each original square face is shared by two cells. Dividing by two gives twenty-four distinct square faces. That agrees with the six orientation families of four squares we counted before.
Carry the method into five dimensions
Here is one edge family of the five-dimensional cube. One coordinate varies and four remain fixed, giving sixteen edges in the family. There are five choices for the varying coordinate, so the entire object has eighty edges. We are showing one direction at a time to keep its sixteen members readable.
Next, one square-face family in five dimensions. Two coordinates vary and three stay fixed. The fixed choices give eight squares. There are ten ways to choose a pair of free directions from five, so ten families of eight give eighty square faces. The equal totals for edges and squares come from different groupings.
Compare another square orientation. It also has eight members. Changing the free pair changes which squares we select, while the three fixed coordinates still supply eight choices. The complete total includes all ten orientation families, not just the family currently shown. This lets us count a large structure without putting every overlapping part on screen at once.
Now select a family of cubic cells. Three coordinates vary, leaving two fixed coordinates and four endpoint combinations. There are ten ways to choose three free directions from five, so the five-dimensional cube has forty cubic cells. Four are visible here. The same method works even when a complete projection becomes difficult to read.
One counting rule, several kinds of part
Return to the tesseract’s square faces, with one orientation family separated. To count a kind of part, make two choices: which directions vary, and where the remaining coordinates are fixed. Here, choosing two directions gives six orientations. Fixing the other two coordinates gives four members in each orientation. Together they account for all twenty-four squares.
Now isolate just one of those squares. It has four vertices and four edges, but it is one two-dimensional face in the larger structure. Mathematicians sometimes use the word face for parts of any dimension: a vertex is a zero-dimensional face, an edge is one-dimensional, and a square is two-dimensional.
Compare a cubic cell. It is a three-dimensional face of the tesseract. In general, a k-dimensional face has k free coordinates. Choose those coordinates from the n available directions, then make two endpoint choices for each remaining coordinate. The resulting count is n choose k, multiplied by two raised to the difference between n and k.
Finish with all eight tesseract cells separated. Try explaining their count without quoting the general expression: one coordinate stays fixed, there are four choices for that coordinate, and either endpoint gives a cell. Then try the same explanation for square faces. The formula is useful because it records this consistent way of choosing directions and positions.