The Geometry of Four Dimensions

  • I will start this development by using analogies and focusing on particular patterns that occur in one, two and three dimensions that also occur in four.

  • The book Flatland by Abbot is excellent both for a geometric point of view and its social satire.

  • This is definitely going to take some time to develop.

If You Lived in a Line.  Imagine that you live in a line...then none of us could deny having a pointy head. More seriously, really, you should try to imagine yourself as a slick slim creature with an eye at both ends, at least then you wouldn't have to worry about seeing your way around when your head is stuck in the wrong place.

For the sake of not being too bored you might as well have a mouth at both ends. In this linear world talking out of both ends will be considered a talent rather than a vice.

Some of you will probably enjoy imagining yourself slithering through your world that is a much longer line. So you are a line segment living in a line world.

Now in this world you can move freely until you hit up against another line segment being. This could get kind of boring if you can't slide by each other, you'd only have two other beings to talk to.

But what if someone becomes territorial ? All it would take is a point being placed in the world to stop your movement !

No worries, you always have your interior life that no one else can see or invade. All anyone else in your line world can see is one of your eyes and one of your bodacious mouths.

But what if something lived in the plane that contains your line ? They could see your insides if they were out in the plane looking in the side of your world. So what is hidden and inside in one dimension is visible and directly accessible from an added second dimension.

Or maybe you are only conscious of being a line living in a line world, but you are actually an elongated rectangle living in a plane ? Maybe you even have unknown eyes or hands at the corners of this rectangle ?

Finally, maybe you have a kind of duplicate line segment, parallel to your line segment body in the line world that can split off and travel anywhere in the plane.

So you used to think I was just a little weird ?

If You Lived in a Plane.  Imagine that you live in a plane...probably a little more fun than living in a line. Think of yourself as some sort of polygon with maybe three, four, five or more sides.

At some place on your polygon you have eyes for seeing and a mouth for communicating.

Now in this world you can move much more freely. Even if you hit up against another polygon you have plenty of room to slide around.

But what if someone becomes territorial ? Now it would take a whole line being placed in the world to stop your movement, even then you would still have a whole half plane to move around in.

In addition to this you always have your interior life that no one else can see or invade. All anyone else in your plane world can see is the outside edges of your polygon.

But what if something lived in the space that contains your plane ? They could see your insides if they were out of the plane looking in the side of your world just like you can see the fuchsia interiors of the polygons on your computer screen ! So what is hidden and inside in two dimensions is visible and directly accessible from an added third dimension.

Or maybe you are only conscious of being a polygon living in a plane world, but you are actually an elongated prism living in a space ? Maybe you even have unknown eyes or hands at other locations in the prism ?

Finally, maybe you have a kind of duplicate polygon, parallel to your polygon body in the plane world that can split off and travel anywhere in the space.

But You've Always Known You Were a Space Cadet.  So do all your friends tell you you're lost in space ? Certainly reading this little treatise will lend credence to their views !

So you are a three dimensional being. At some place on your body you have eyes for seeing and a mouth for communicating.

Now in this space world you can move much more freely. Even if you hit up against another person you have plenty of room to slide around.

But what if someone becomes territorial ? Now it would take a whole plane or very high fence being placed in the world to stop your movement, even then you would still have a whole half space to move around in.

In addition to this you always have your interior life that no one else can see or invade. All anyone else in your space world can see is the outside edges of your body.

But what if something lived in the four-dimensional space that contains your space ? They could see your insides if they were out of the space looking in the side of your world ! So what is hidden and inside in three dimensions is visible and directly accessible from an added fourth dimension.

Or maybe you are only conscious of being a body living in a space, but you are actually a higher dimensional being? Maybe you even have unknown eyes or hands at other locations of your whole four-dimensional being ?

Finally, maybe you have a kind of duplicate body, parallel to your corporeal body in the space world that can split off and travel anywhere in the four-dimensional space.