FINALLY! A Good Visualization of Higher Dimensions

in this video we will look at a geometrical problem in many dimensions but don't be scared there will be no formulas and we will start in two dimensions let's create a square with side length one inside the square we place four blue circles one towards each corner they have diameter one-half so they fit exactly inside the square this leaves an empty region in the center where we place an inner circle we are interested in the size of the inner circle we will consider generalizations of this figure in three four and more dimensions and we will be interested in the size of the inner circle or hyposphere as it is called in higher number of dimensions the question we ask is what happens to the size of the inner hyposphere as the number of dimensions grow the surprising answer is that the inner hyposphere grows without bond already in 10 dimensions it is so big that it stretches outside of the hypercube that contains the blue hypospheres in the two-dimensional case an inner circle of that size would have to overlap with the blue circles but we don't allow any overlap so how can this happen in 10 dimensions how can the inner hyposphere extend outside the hypercube when it is also between the blue hypospheres which are all inside of the hypercube a mathematician may explain it using pure equations an alien might be able to visualize it directly in 10 dimensions an artist may draw a crazy interpretation like this illustrating the strange fact that the red hyposphere extends outside the boundaries but crazy as it is this figure is not based on the underlying math and it doesn't explain anything let's look at the situation in three dimensions we create a cube with side length one inside the cube we put eight blue spheres with diameter one half one towards each corner in the empty space in the center we put an inner sphere that touches the blue spheres how big is the inner sphere in this 3d case compared to the 2d case can we understand this visually to do this we make a two-dimensional cut through the cube like this the cut follows two opposite edges of the cube and goes through its center the cut only passes through four of the blue spheres and they appear as circles the inner sphere also appears as a circle in the cut note that the cut of the 3d figure is somewhat similar to the 2d case but there is an important difference the cut creates a rectangle which is stretched out such that there is more space between the blue circles in one direction this extra space allows the inner sphere to be bigger in the 3d case compared to the 2d case what happens in 4 dimensions it is difficult to visualize 4 dimensions but we can make a two dimensional cut in much the same way as we did in 3d and look at that just as in the 3d case the cut follows two opposite edges of the 4d hypercube and goes through its center in this cut four of the blue hyperspheres are visible as well as the inner hyposphere so the cut looks much like the cut from 3d but it is stretched a bit more this is because it contains the diagonal of the hypercube which is longer than the diagonal in the 3d cube because of this additional stretching there is even more space in the middle allowing the inner hyposphere to be even bigger the same technique works in any number of dimensions let's look at the 2d cut of the 10-dimensional figure the hypercube appears as a stretched rectangle in the cut it is stretched quite a lot since it contains the hyper diagonal of the 10-dimensional cube four of the blue hyperspheres show up as a circle in the cut note how far apart they are spaced in the diagonal direction of the hypercube finally we can see the size of the inner hyposphere how it extends outside of the boundaries of the hypercube we can see why this happens it is because there is a lot of room between the blue hypospheres in the diagonal direction what can we learn from this a direct learning is that diagonals are longer than sides and that this effect becomes stronger the more dimensions there are the longer distance along the diagonal comes because you have to travel along many dimensions simultaneously perhaps a more interesting learning is that it can be possible to visualize high dimensional geometry if you look at them the right way thanks for watching see you next time

Channel: Benjamin Wiberg

Views: 147,724

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Id: sZqGWy0hxe8

Length: 5min 1sec (301 seconds)

Published: Sun Aug 22 2021