It goes by many names: the hypercube, the 8-cell, or the octochoron. It is represented by many shapes; a small cube inside a larger cube, two cubes connected by a bridge, a cube with slightly skewed angles. All the tesseract's names and shapes are attempts to visualize what we can never understand. What does the fourth dimension look like?
Most people's first encounter with the word 'tesseract' was in the dimension-hopping parts of A Wrinkle in Time. Madeleine L'Engle's characters hopped through vast reaches of space by 'folding' space into higher dimensions, bringing the edges of the three dimensional universe together the way someone might fold the edges of a towel together. This practice was called 'tesseracting,' and it worked very well, until the characters were nearly killed by accidently getting into a two-dimensional universe. L'Engle wasn't making the term up. She used a term coined by mathematician Charles Howard Hinton in the late 1800s. It refers to a cube, or what a four-dimensional being playing with four dimensional objects would consider a cube.
Making a Mental Tesseract
Picturing things in four dimensions is not physically possible for provincial three-dimensional consciousnesses such as ours. It's possible to envision, however, a building process that lets us understand the general idea of a tesseract, or hypercube. A point, a one dimensional object, has one 'corner' and nothing else. A line segment has two corners, and length, but not width. A square has four corners and length and width, but no volume. A cube has eight corners, length, width, and volume. Building on this, we understand that a hypercube, or tesseract, has to have sixteen corners (as the number of corners doubles each time), and length, width, volume, and some new quality.
If we want to visualize the making of a tesseract, we would again start with a point. If that point were made of malleable putty, and someone put their finger on it and drew it out in one direction, it would be a line segment. If that same person put their fingers on the corners of the line segment, and pulled them out, perpendicular to the line segment, the putty would form a square. If they were to take the four corners of the square and draw them out, perpendicular to the edges of the square, the putty would stretch into a cube. To form a tesseract, these magic fingers would have to swoop in again, take hold of each of the corners of the cube, and pull them out in a direction perpendicular to all the edges of the cube.
In three dimensions, of course, that last part is impossible. But add another dimension, and it could happen. This is why so many pictures of tesseracts show a smaller cube inside a larger one. That is the best visual approximation we have of a cube being pulled outwards. But it's not an actual tesseract. Just like each of the faces of a cube is a square, each of the faces of a tesseract is a cube.
Playing With Four Dimensions
The tesseract has tantalized people since its conception. Most children are pleased when they find that they can draw simple cubes on a piece of paper by connecting two squares with straight lines. It's a mental breakthrough, making three dimensional 'shapes' out of two dimensional lines. The tesseract seems to require nothing more than the same mental breakthrough. Just take those cubes and connect them . . . somehow.
Hinton himself, after writing a book about tesseracts and the fourth dimension, marketed 'magic cubes' that he claimed could help people see, or understand, the fourth dimension. This being the great age of Victorian spiritualism, they were used during seances to allow people to see ghosts, who, it was supposed, were as likely to hang out in the fourth dimension as anywhere else.
Another great way to dabble in the fourth dimension is the Rubik's Tesseract. It could be marketed to those who don't find the Rubik's Cube maddening enough, or the families of obnoxious people who wanted some peace and quiet. When playing with a Rubik's Tesseract, one would not move it so that the faces matched up, but so whole, complicated volumes matched up. It is calculated that the Rubik's Tesseract would have 1.76 x 10^120 possible positions.
We're Gonna Need a Bigger Vocabulary
When people wade in four dimensions, there are practical problems that arise. One of them is the same thing that trips up time travelers; semantics. Suddenly we are dealing with problems that do not exist, except on the ragged edge of the hypothetical. Time travelers need a verb tense to explain that something will happen in their personal future but the entire world's past. Dimensionologists (and I sincerely hope they call themselves that) need words to explain moving as a point, a line, or a volume, through four dimensional space.
Perhaps the best example of this need for new words is the perfect magic tesseract. Magic squares are grids that, when properly filled with numbers one to N squared, have the same sum when the numbers in a row, column, or a diagonal are added up. There are also magic cubes, which go from one to N^3, and the rows, columns, diagonals, and pillars all have to add up. And then there is the magic tesseract, in which the numbers range from one to N^4, and everything has to add up again, except there's an extra dimension in there. In order to explain that the numbers for this dimension need to add up to the same sum as every other row, mathematicians added the name 'file' to express the numbers lined up along another dimension. They also had to chuck the word 'diagonal,' since diagonals don't quite work, substituting the word quadragonals for the lines of numbers that passed through the center of the object and joined the corners. In a magic tesseract, the rows, columns, quadragonals, and files all have to add up.
And then there is the fresh hell of a perfect magic tesseract. In this fabled creatures, the rows, columns, pillars, files, quadragonals, need to add up to one number, as do the diagonals of the cubes. In other words, if someone were to pick out one cube of the perfect magic tesseract, it's rows, columns, diagonals (both on the faces of the cube an through the center) would all have to add up to this number. If some one were to then take one face of the cube, its rows, columns, and diagonals would have to add up to the same number. Believe it or not, it has been done. A Perfect Magic Tesseract was constructed, using the numbers one to 65,536, or 16 squared. The sum of all lines, everwhere, was 534,296. This feat was achieved by John Hendricks in 1999. We can only assume that Hendricks was then sucked, brain first, into a higher dimension, where he is probably in a bar, still bragging.