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The Idea
In the Maze
> Notes
Reference

The Fourth Dimension
Can You See It?
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The Tesseract
The Tesseract, Part Two
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How Much Space Is There?
Rotations
How to Point
How to Orient Yourself
 > Volumes
Walls Are Opaque
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Some Mathematics
Bibliography

## Volumes

What I want to talk about here is the concept of volumes. The concept is implicit in all the pages of the guided tour that deal with passages, notably the one about the forward passage, but somehow it never came through as clearly as I wanted it to. So, let's go back and work through one of the examples again.

The example I have in mind is the first of the forward side passages, the one that went forward and then right. To construct it, we started with a forward passage.

Then we removed the right wall and saw the back wall of the new square. Let's not do that yet, though. Instead, let's look more closely at that right wall.

Now, as I said back when I was explaining the idea, the three-dimensional retina really ought to be displayed as a solid block of color. In particular, the wall we're looking at really ought to be a solid block of purple.

Now that we know the wall is a tangible three-dimensional object, let's remove it.

That empty space is an example of the kind of volume I'm talking about.

Finally, we can see what's behind the right wall. The back wall of the new square is in the middle; around it are parts of four other walls (up, down, in, and out).

The spaces that those walls and parts of walls occupy are also volumes.

Now maybe the following generalities will make some sense.

• A volume is a space that can be occupied by a wall, or by the visible part of a wall.
• Any volume can be subdivided into smaller volumes.

Let's call a volume unoccupied if it is not occupied by the wall that defines it.

• A volume can be unoccupied “from above”. If there were no forward passage, for example, the right-wall volume would be covered up by the larger wall immediately in front of us.
• A volume can be unoccupied “from below”. That's what happened to the right-wall volume when we removed the right wall and looked behind it.

A good way to make yourself more aware of volumes is to use the change keys to mess around with the render depth.

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