Visualizing the Fourth Dimension Quiz

Answer: sixteen

The pattern here is just that the number of corners (or "vertices") doubles each time we go up a dimension. So if you believe the pattern continues, you know what comes next without even having to look at the fourth dimension!

To see where the pattern comes from, it's again easiest to visualize in lower dimensions. The square consists of a top line segment and a bottom line segment connected to each other. The cube consists of a top square and a bottom square connected to each other, so has the same number of corners as two squares. By analogy, we expect this mysterious hypercube to consist of two cubes connected to each other, so to have twice the number of vertices of a cube.

Answer: eight

Again, we can guess at the answer just by looking at the pattern from the lower dimensional (easier to visualize) boxes -- once you see 2, 4, 6, ___, you might guess the blank is filled by 8.

Mathematically, the explanation for this pattern comes from the structure mentioned in the answer to question 1. Just like the six faces of a cube (or the six sides of a die) correspond to the bottom, the top, and one face extending upwards from each side of the bottom square, the eight faces of a hypercube consist of the top, the bottom, and one face extending upwards from each side of the "bottom" cube.

Answer: thirty-two

The difficulty here is that it's hard to see anything from the numerical pattern. Even if we include the one dimensional "box" (that is to say, a line segment), what number should come next in the sequence 1, 4, 12, _____ ?

The key is to focus not just on the "what" but on the "why" as well. Why were there twelve edges in a cube? Because we added up the edges on top, the edges on bottom, and the edges connecting them. For a hypercube, there are twelve edges on top (remember that the top layer of a 4-dimensional hypercube is a 3-dimensional cube) and twelve on bottom. Just like an edge extends downwards from each of the four corners of the square on top of a cube, an edge extends downwards from each of the EIGHT corners on the cube on top of a hypercube. Adding twelve, twelve, and eight makes thirty-two.

Answer: Multiply it by sixteen

As usual, this can be explained by looking for patterns in the lower dimensions. The area of a rectangle is length times width, and that of a three dimensional box is length times width times height. Looking at this, its natural to think that the area of a box in four dimensions is the product of its lengths in each of the four directions. If we multiply each direction by 2, we multiply by 2 a total of four times, and 2 times 2 times 2 times 2 is 16.

So to summarize what we know, this mysterious "hypercube" is a four-dimensional object with sixteen corners connected together by thirty-two edges. It can be thought of as being made either by attaching together eight 3-dimensional cubes or by taking two 3-dimensional cubes and connecting the corners of one to the corners of the other. Not a bad description for people who only live in three dimensions!

Answer: 2

In 1 dimension, the two opposite "corners" (endpoints) of a line segment of length 1 are (by definition) 1 unit apart from each other. In two dimensions, the diagonal of the square is sqrt(2) (for example by the Pythagorean theorem -- the diagonal is the hypotenuse of a right triangle where the other two sides are both 1). In 3 dimensions the diagonal of a cube is sqrt(3). So we have 1, sqrt(2), sqrt(3),... the next term is sqrt(4), which is 2.

More generally, there's a distance formula in higher dimensions that works similarly to the Pythagorean theorem, or to the 2-dimensional distance formula you may have learned in geometry: If you have two points (x_1, x_2, ..., x_n) and (y_1, y_2, ..., y_n), then the distance between them satisfies

distance^2 = (x_1-y_1)^2+(x_2-y_2)^2+...+(x_n-y_n)^2

Here n=4 and the two opposite corners are (0,0,0,0) and (1,1,1,1), so the distance^2 is 1+1+1+1=4.

Answer: Five

A triangle has three vertices. A tetrahedron has four (the three from the triangle and the new one), and the simplex adds one more. Each time you go up a dimension you add a vertex.

Simplices show up in practice in many resource allocation problems. For example, suppose you have $100 to divide up among five people. If you look at all the different ways of dividing up all the money and plot them, the shape ends up being a 4-dimensional simplex. The five vertices of the simplex then correspond to the most extreme possible distributions -- one person out of the five gets all $100.

Answer: Five

A 1-dimensional line segment has two 0-dimensional endpoints. A 2-dimensional triangle has three 1-dimensional sides. A 4-dimensional tetrahedron has four 2-dimensional faces. Two, three, four... it looks like this time the vertices and faces follow exactly the same pattern.

One way of thinking about this: In a triangle, each vertex has a side opposite it. In a tetrahedron, each vertex has a face opposite it. Similarly, for each vertex of a simplex, there's exactly one face opposite it that contains all the vertices EXCEPT that one vertex. So there's a one-to-one correspondence between vertices and faces now.

Answer: Ten

One way of seeing this: Recall that we can think of a 4-dimensional simplex as a tetrahedron, along with a new fifth vertex connected to it. So the simplex will have all the (six) edges from the tetrahedron, plus some new edges coming from the fifth vertex. That vertex has four other vertices to connect to, so gives four additional edges.

The pattern here is that of the "triangular" numbers: 1,3,6,10,15,21... The difference between consecutive numbers increases by 1 each time: 3 is two more than 1, 6 is three more than 3, 10 is four more than 6, and so on.

Answer: Multiply it by sixteen

This is the exact same answer as for the 4-dimensional hypercube -- the number "sixteen" here is a property of the dimension, and not of any particular shape we're talking about. If we take any 4-dimensional object whatsoever and scale it up by a factor of two in every direction, the volume will increase by a factor of sixteen.

One way to think about this is to imagine scaling it up one direction at a time. Each time we stretch it by a factor of two in a single direction, we're also multiplying the volume by two (if we make an object twice as long, we've doubled it's total volume as well). We're in 4 dimensions, so we need to perform this scaling 4 times. And multiplying by two 4 times in a row is the same thing as multiplying by sixteen.

Answer: Flatland

In the story, A. Square first dreams of Lineland, a 1-dimensional world where the inhabitants all live along a single line. He tries to convince the points there that there really is a second dimension, but the points can only see their own dimension, and don't believe him. He then awakens in his own 2-dimensional world, and is visited by a 3-dimensional sphere. The sphere tries to convince him that there really is a third dimension, but he can only see what happens in his own 2 dimensions, so doesn't believe the sphere at first.

Once he finally understands, his mind starts to work along the same analogies as were in the earlier questions in this quiz -- if there's a 3rd dimension orthogonal to the two that he knows, why should there not be a 4th dimension orthogonal to the one the sphere (and Abbott's readers) live in?