## 4D Tennis

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

### 4D Tennis

After 20 years I finally figured out how to to visualize this (as though it were a computer game). I sure feel stupid it took so long.

In our world the surface of a tennis court is a 2D rectangle. In a 4D universe I believe it would be a 3D cylinder. We draw the cylinder by drawing the ana-kata dimension in place of what is traditionally the up-down direction. Next, imagine there is a light directly above the court and this casts a shadow beneath the ball. That shadow is always either on the surface that is the cylinder or close by (out of bounds). Finally, draw the ball too. It is always directly above the shadow. The distance between the ball and the shadow is naturally how far the ball is above the surface. The ball bounces or rolls whenever the ball and shadow merge. One dimension in the drawing is shared between two of the four dimensions. They are presented differently and thusly may be distinguished, easily thinks I. Same thing with the tennis racquet. Show both the racquet and its shadow. This should work for any game with an airborne object -- golf, football, baseball, discus, etc.

It is possible to tell when the ball crosses the plane of the net by looking at the shadow. If it is on the other side of the net then one sees the shadow through the cords of the net. When the shadow comes to our side then we artificially have it obscure the net cords. (This wouldn't really happen, but that's a quibble.) If the ball is high enough at that moment then it passes over the net.

What we are doing is using the viewers knowledge of the situation to show all four dimensions using only the three available to us. We are using one dimension in our drawing in two ways, thus showing two dimensions of 4D.

One of the most bizarre things about 4D is that a person could rotate 360 degrees while still facing forward and standing upright. This is one of those things I cannot yet imagine, but in this situation we dont have to show it. Instead take advantage of this being point-of-view and rotate the court. Note that if the players' bodies really allow this, then during the serve there is no effective difference between the left and right hand sides of the court. When switching from one service court to the other simply rotate to one's preferred orientation. In general, in order to use one's preference of forehand or backhand it isn't necessary to run past the ball. The player could rotate instead.
PatrickPowers
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### Re: 4D Tennis

PatrickPowers wrote:[...]One of the most bizarre things about 4D is that a person could rotate 360 degrees while still facing forward and standing upright. This is one of those things I cannot yet imagine, but in this situation we dont have to show it. Instead take advantage of this being point-of-view and rotate the court. Note that if the players' bodies really allow this, then during the serve there is no effective difference between the left and right hand sides of the court. When switching from one service court to the other simply rotate to one's preferred orientation. In general, in order to use one's preference of forehand or backhand it isn't necessary to run past the ball. The player could rotate instead.

This is actually very easy to imagine if you use my projection-based approach to 4D visualization. If you imagine a hypothetical 4D person facing you, his face approximately projects to a sphere. Down the middle of the sphere (on the inside) is the conical projection of a nose. Below this is a pair of flat half-ellipsoids -- projections of the lips, and two small spheres on opposite sides of the nose cone where the eyes are. (Let's ignore for the moment that an actual 4D being would probably have more than 2 eyes -- the dimensional analogy is clearer without getting into that complication.)

Now imagine the different ways this sphere can rotate. There are 3 planes of rotation involving the 4th dimension that changes the direction the 4D person is looking at -- let's disregard those for the time being. This leaves 3 other planes of rotation -- the 3D ones -- that leaves the face facing the same direction. There are 2 rotations that would tilt the head sideways: the nose cone will tip over in either of the 2 horizontal dimensions: these are analogous to us 3D beings tilting our head sideways.

The last rotation is the one you refer to: this involves rotating the face around the vertical axis, but not involving the 4th dimension. This spins the two eyes around the nose cone but preserves the vertical orientation of the face. The face continues to face forwards in 4D (the forward direction projects to the zero vector) and remains upright, but the 4D person is actually turning his head around. The overall effect from your POV is that his face spins around the horizontal plane (in projection).

My speculation is that this kind of motion could be a 4D-specific gesture akin to a shrug, or maybe a muted happy dance. Or a playful gesture. 4D faces can be a lot more expressive than a 3D face!
quickfur
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