Projecting 4D.

[wendy]

Let's look at 4D, not so much from a POV, but a rather interesting eyepeice.

This particular 4camera is something like an iPad, but it acts like a camera. Rays of light are filtered so that only perpendicular light is admitted, so when you point it as something, the image is only as large as the screen. Mathematically, the rays start at an object, travel through clear material (air, say), and fall perpendicularly on the screen, is all the device captures.

We get a capture of its screen in a projector, a clear box, whose every solid pixel matches a solid pixel on the screen. We can turn the object around like a glass cube, much as we might turn photos and maps around. We can use some magic to remove and restore objects from view. Otherwise, we might not see every pixel that is presented.

What we see in the projector is a still of what the 4camera is seeing. We see geometrically, at x,y,z, the first non-clear point at +w,x,y,z of the 4camera.

Tesseract, face first

We now point the 4camera at a tesseract, so that its edges are parallel to the rays. All we see in the screen is a cube. Although we suppose the cube to have inside and out, all of the rays have arrived from the outside of the tesseract, so our illusion of inside and out is just that: an illusion. We can turn our box around, and see the six squares. This is where the six adjacent cubes join the cube we're looking at. Likwise, because we have not moved the camera in 4D, what we see is what we would take to be a cube with six faces, but this is entirely a surface of the tesseract.

We look through our collection of photos of the cube, and find a cube square-first. We see the body of the square representing the first face, and the four edges representing the four squares pointing outwards.

Placing this image progressively on the six squares of the cube in the box-camera, we see that they are exact matches. The six faces of the tesseract exactly match the cube face first.

Tesseract, square-first

We now turn the tesseract so that the thing is square-first. The projector shows a 1:1:sqrt(2) image. There are two squares at the ends of this are still squares. These are the cubes, squar-first, presented from an edge. There are also, four stubby prisms. We see essentially a 1:1:sqrt(2) prism, divided in the middle of the sqrt(2) sides. The square at the middle is the closest to us. Above and below this, there are stubby cubes. The four sides represent 1:sqrt(2) prisms, and the top and bottom are squares.

The squares top and bottom are in fact just squares. This is where one of the sloping cubes joins its mate just over that square. It's a full-on square, but nothing like what we saw before. The four tall sides are things we can identify from our collection of photos as an edge-first cube. We put these images on, and there's a perfect match.

To get the stubby prisms, we have to slope our face-first cube to 45 degrees. This makes the height smaller, but keeps the base. We get a perfect match here too.

Tesseract, edge first

We now move the 4-camera around, so it's showing an edge-first view of the tesseract. The image we're presented with in the projector is a hexagonal prism, but we can use our thing to show that there are several items of note.

The hexagons match the photograph we have of a vertex-first cube. But we see also that the six rhombs that make it up (twice) would represent us looking through it. Only three of these rhombs are made into prisms on the facing side. This is because the 4-camera can't see through the tesseract, and only the facing side is apparent in the image.

The rhombs themselves are edge-first cubes, tilted so that the top of these appear to take a 120-degree rhombus. Three of these rhombs sit around a line, which we see really is a line. The squares between the three facing cubes are squashed as rectangles, the squares between the rhombic prisms and the hexagon ends appear as three of the six rhombs.

Tesseract, vertex first

We now move the 4camera again, so that it's looking at the vertex-first presentation. The general shape of the image in the projector is a rhombic dodecahedron, but we can untangle from this, four separate and somewhat squashed cubes. These cubes have been forshortened along their long axis, so that axis is the same length as the edge. The rhombs are 109 degrees, which appear to us to be oblate.

No image in our collection of photos match these images, but we know from our photos of real things, that when something is at an angle, it is forshortened in the direction of slope. The thing, when stretched out, become cubes. We see four cubes, and the twelve rhombs that form the rhombic dodecahedron form twelve of the twenty-four squares between the cubes. These twelve link up to the four non-visible cubes. There are six on the facing side, between pairs of rhombohedricly projected cubes.

Pictures and Maps

We can now point our 4camera at printed four-matter. These are pages of a book, say, which to the 4D eye, are exactly like our 4-camera. The 4-d being can render every point of the 3-space, as we might 2-space.

For our point, we should remember the main difference between a picture and a map is that things can fall in a picture, but don't fall in maps. In a picture, object in the top left-hand corner will fall to the bottom because of gravity. On a map, gravity is perpendicular to the map, so things don't fall to any side of the map.

This is a useful point, because our projector is always in the field of gravity, and unless we suspect otherwise, we might suppose our gravity is also present in the image, too.

The 4being has presented us with a plan of his house, on a clear transparency. On this the walls are lined in black, say. We see in the projector, a series of various boxes or prisms, all of which are solid around, but clear inside. Between the prisms, there are openings, randomly placed on the six walls. These are doorways. There might be other kinds of markings on the walls that represent windows. In our plans we represent these as a box with a cross on it.

The 4being now puts on a mass of little 3d prisms of various shapes and sizes. This is what we know about furniture. What he is going to do is much like what we do: put various cupboards and beds and so forth up against walls and in rooms. To not obscure windows and walkways. The bits of furniture are presently sorted out, and we see them arranged up against various walls of the cells, with little regard for what we take as gravity. It is a plan, the gravity is lost in the w direction of the 4camera, and anything that falls, lands and stays, save for some rolling.

Pictures work the same way. Let's watch some coins, of various shapes fall. We put the 4camera so it's standing upright. That means, for our prospective, something tossed will fall from +z to 0. A handfull of disks of various shapes, money from all nations, we might suppose, is tossed in at the 4top, and let fall. The fall is chaotic, but at the end, the xy plane of the projector is filled with little prisms, all with their minimal height in the z direction.

Our examination of these prisms show that they look like they had been cut from a thin sheet, say of coolite. The shape of the bases are similar to what we see as various polyhedra, from random angles. Because the image is now still, we are unable to rotate the polyhedra, so we might, for example, see a dodecahedron, not from a symmetric view, but just some random angle. The dimension gained in our vision is height, here mostly empty, and remainderingly taken by the heights or thicknesses of the disks. The xy dimension holds pictures of polyhedra, exactly as we might have seen them in photos, but we are unable to alter w ourselves, so we can not rotate the polyhedra, except to rotate the projector.

A 4being had no need to resort to this knavity, since his eye, like the 4camera, is perfectly capable of resolving the colour of the first non-clear object in a line. Since this might pass through a full 3space that might form his glasses, we can see that each ray would, fall on his glasses on an xyz coordinate, as much as it does on our camera. The retina mirrors the lens of the glasses, is also 3space. Unlike our 4camera, the relevant rays are not perpendicular, but converge on the iris, and fall on a 3d retina that backs his eye.

Like us, the 4being needs to process depth, but we use learnt cues to do this. And yes, they are learnt.

[wendy]

We now move onto the second element.

The regular convention is for the circle to follow the clock (clockwise). One starts at the top (12) , and proceeds through 1, 2, 3.. The mathematical convention goes from 3 oclock, and goes anticlockwise, ie 3, 2, 1, 12, 11. This moves x -> y -> -x -> -y -> x, or 1 to i to -1 to -i to 1. Swapping the x and y axes, makes the circle go clockwise (12, 1, 2, 3, ...).

When we couple Anticlockwise with 'Up', and 'clockwise' with 'down', we get the helix of a common screw. Turning the screw clockwise, makes it go down or into the wood, turn it anticlockwise makes it go up, or out of the wood.

The directions x,y,z form Ac, U when turning a screw from x=1 to y=1, makes it advance in the positive z direction.

Any change of a pair of coordinate, or swapping Ac/Cw, or swapping U/D, gives a reflection. Two reflections gives the original.

In four dimensions, we might represent space, not as 1+3 d, but as 2+2d.

The parity of rotations is not based on a screw as in 3D, but on what is called a 'clifford rotation'. When seen from 2D + 2D, we see the rotations are of one kind, if they are both clockwise or both anticlockwise, and of the other if there is one of each. One can not bring one to another except through a rotation.

If the rotations are the same speed, a truly intresting and unexpected result happens. All of the points go around the centre of the 4sphere, there is only a single stationary point. You can quite easily see this if you are versed in complex numbers.

In 4D, one has four axies (w,x,y,z), which we might present in two views (w,x), and (y,z). There is a point in 4space for every pairing of points from each 2space. When we fix w,x, and let y,z freely vary, then we see that the point w,x on the left, represents an entire plane, and likewise any point y,z represents the whole hedrix w,x. It is therefore not appropriate to call the 2-space a plane, since it no longer divides space. Instead, the division of space is a single equity in a single side, represents a 3space. A 3-space, might be represented as a line on one side, and the whole of the 2-space on the other side. One can not cross the 3space without going through it.

The space CE2 is Complex Euclidean Space, or euclidean space with complex numbers. The same analytical formulae that serves the regular 2-space still work, but the coordinates are complex numbers, and the lines become argand diagrams, complete with a sense of rotation.

A line, for example, is represented by Y = a.X + b, where a is the slope and b the intercept. We can readily set b=0 for this exercise, to give Y = a.X. In such a mapping, we see that things like dialations of the form X, Y => vX, vY will preserve the alignments vY = a.vX => Y = a.X for all X, Y. In complex numbers, we can consider vY = a.vX = av.X includes the instance when v = cis(wt) = exp(iwt), where w is a speed constant. What this means, that apart from resizing, we have a full rotation of every point in 4space, without changing the intrinsic slope Y/X of any point.

This means, in the simplest case, there is a series of 'right-parallels' or 'left-parallels', which given a single 2space (with a rotation thereon), and a point not on this 2space, there is a single 2space that passes through the point, and is right-parallel, and a different one that passes through that is left-parallel.

It is the nature of physics, that a body that has several modes of rotation, will by tidal effects, etc, tend to equalise the energy of these. A perfectly 4sphere, of uniform density, will by tidal effects, tend to adopt a clifford-rotation of either left or right sense, the sense will generally agree with the star it orbits, in much the same way that planets in 3space tend to adopt a rotation that is pointing in the same general direction as its orbit, and as that of the sun. For our system, the right-hand rule, which says that land is pushed in the direction of the fingers of the right hand, the thumb pushes north.