After this second step, our calculations are telling us that this is a 1.38-dimensional graph, but we can clearly see that it’s a two-dimensional graph.Ĭloser, but still not right. Well, that’s a bit better, but it’s still wrong. Let’s move another edge in every direction: All that work, and our calculations tell us that this is a zero-dimensional graph? That’s obviously not right. I’m going to start again from the node in the middle of the graph, and every time we move one more edge in every direction, I’m going to calculate D, the dimensionality of the graph. So now we can measure the dimensionality of our simple graph. If you’re interested in the formula I use to calculate D, check out the footnote below. I’m going to call that number D, the dimensionality of the graph. I’m going to spare you the mathematics, but every time we move one more edge, we can calculate whether V is increasing with r to the power of 2, or 3 or some other number. Well, we’re going to do the same thing the crabs did, to find out whether this graph is two-dimensional or three-dimensional. Remember how the two-dimensional crab found that the amount of space it covered increased with the distance it moved to the power of 2, and the three-dimensional crab found that it increased with the distance it moved to the power of 3? We can keep doing this, moving 3, 4 and 5 edges in every possible direction: By doing so, we reach 13 nodes: the 5 we reached within 1 edge, now shown in darker red, plus another 8 we reach within 2 edges, shown in brighter red: Now let’s go two edges in every possible direction. So by moving 1 edge in every possible direction, we’ve covered 5 nodes’ worth of space: In this case, we’ve covered five nodes’ worth of space, including the one we started from and the four nodes we’ve just reached: In the graph, however, we always reach a whole number of nodes. The amount of space the crab covered was measured in numbers of crabs, including fractions of a crab: the crab might cover 3.14 crab’s worth of space, or 12.57 crab’s worth of space. The only obvious measure of the amount of space we’ve covered in our graph is the number of nodes we’ve reached.Īgain, you can see how a graph is different from the crab’s continuous universe. The only obvious measure of the amount of space the crab covered in its universe was the amount of space the crab took up itself. Next, we need to decide how much space we’ve just covered. In this graph, however, we can only move in four directions from that middle node, along the four edges shown in red. The crab was able to move in an infinite number of directions, tracing out a circle. You can see how a graph is different from the crab’s continuous universe. So the only way to measure distance in the graph is to count numbers of edges.įollowing the crab’s example, we’re going to move one edge in every possible direction: But I could have drawn all the edges different lengths, or I could have drawn them half a micron long or half a mile long, and it’d still be the same graph. I mean, I’ve drawn all the edges of this graph the same length, about half an inch long, depending on what kind of screen you’re looking at right now. Remember that the edges of the graph don’t have any particular length. The only obvious measure of distance in our graph is one edge. The only obvious measure of distance in the crab’s universe was one crab length. To use the crab’s method, first, we need to decide the distance we’re going to move. Remember that our crab determined the dimensionality of its universe by measuring how much space it covered moving different distances in every possible direction. Let’s start from the node in the middle of the graph: Now, I’m pretty sure I know the dimensionality of this graph.įrom our God’s-eye perspective, that’s pretty obvious.īut let’s do this from a crab’s-eye perspective, from inside the graph. Here’s a hint: it’s not two and it’s not three. I’m finally going to answer the question: how many dimensions are there in this universe? Now I’m going to use the same crabby method to determine the dimensionality of graphs generated by Wolfram Physics. In my article How to measure the dimensionality of the universe, I introduced a mathematically-minded crab, which was able to determine the dimensionality of its universe by measuring how much space it covered moving different distances in every possible direction.
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