The Hypercube Revealed |
Rotations and Projections |
Tips and Suggestions |

The best place to start exploring 4-dimensional space is with thehypercube(or4-cube,tesseract,octachoron). And the best way to understand the hypercube is by analogy with its 3-dimensional version, the 3-cube.The viewer shows a 4-cube by default. We'll also use it to show a 3-cube and draw parallels between the two. (You can follow on two viewers by opening the viewer page in two browser windows; to display a 3-cube, click

polytopes..., then in the polytopes dialog select dimension 3 and clickOK). In both windows, clickreset polytopeand make surew-perspectiveandz-perspectiveare unchecked. Both viewers should show:

cube initial view, 3-d and 4-d In

ndimensions, ann-cube's edges run innperpendicular directions. For each cube in its initial position, the first direction,x, is horizontal, the second,y, is vertical, and the rest are perpendicular to the plane of the screen, so without perspective they are hidden. Rotating each cube slightly in thexy-plane by pressing thexybutton gives something like this (to reverse rotate, right- or Ctrl-click):

xy-rotated cube, 3-d and 4-dSince the rotation only involves

xandy, the edges perpendicular to the screen remain so, still hidden. Now pressxzto rotate each cube horizontally, getting something like this:

after xz-rotation, 3-d and 4-dNow edges are seen running in three directions. For the actual cube edges, these directions are all perpendicular, but since the screen is two-dimensional, their projections cannot be. After the

xz-rotation, the edges that were formerly aligned in thezdirection now have an "xcomponent" and are seen on the screen as horizontal.Notice that the 3-cube can be seen as the two squares that were previously superimposed now displaced and with corresponding corners (

vertices) joined:

opposite faces of 3-cube (To color the faces of the 3-cube on your viewer, click the

cells...button, undernew cellin the cells dialog clickbegin, and click on the edges of the face you want until it's colored (clickclearif you get the wrong face), then clickfinish. Clickbeginagain to highlight another face, clickclear allto start all over. See here for an explanation of this facility. Or you can just do this. Note that you can't do this reliably with the 4-cube at this point, since every edge you see is really two, with the samex,y, andzvalues, but differentw(see below), and the program doesn't know which you mean to click on.)A 3-cube in fact has three pairs of opposite faces thus connected, the other two being:

The 3-cube's story ends there, but the 4-cube, which you should have looking pretty much like the above without the colors, still has a set of edges hidden, running in a fourth direction I label

w. To see them, rotate the 4-cube withywto give them aycomponent:

4-cube after yz-rotationNow you're seeing the entire 4-cube. To make some sense of this confusing figure, notice that

itcan be seen as two copies of the 3-cube we were just seeing, that were superimposed and are now displaced vertically and connected by the newly-visiblew-edges:

4-cube with opposite facets highlighted Analogous to the 3-cube and its square faces, the 4-cube has four pairs of opposite 3-cube "facets". They do the same job, forming the boundary of the four-cube, separating its interior and exterior. It's a good exercise to find the other three pairs yourself (use

cells...!), or look here.

To study four-dimensional polytopes that live in

3-d perpendicular projection onto xyplane

xyzw-space, weprojectthem down to familiarxyz-space, and then project that three-dimensional figure onto thexy-space of the viewscreen. With perspective turned off, the projection is perpendicular, as above in the 3d-to-2d case. An important point is that rotations that involve only the dimensions of the space we're projecting to don't essentially change the projected figure - they only rotate it. For 3d-to-2d this is particularly simple:

xy-rotation gives a congruentxyprojectionWe'll take the same idea one dimension higher. To get a better sense of how the

xyz-projection sits in 3-space, clickoptions..., then under bothpolytope exteriorandpolytope interiorcheckz-shading, and underpolytope interiorpressdash. Working with the default 4-cube, starting fromreset alland pressingxy,xz,yz, andxwbriefly will give something like this:

Consider that the 4-cube has been perpendicularly projected to our familiar

xyz-space, wherexandyare the respective horizontal and vertical dimensions of the screen andzruns from you toward the screen. Edges are shaded more dimly the farther they are from you in thezdirection, and edges in the interior of thexyz-projection are dashed (in 4-space itself, the 4-cube has no interior edges, any more than a 3-cube in 3-space does). Now try any ofxy,xz, and/oryz, and notice that you have essentially the same 3-d figure. Just as with the case of 3d-to-2d projection, applying rotations that involve onlyx,y, andzto a 4-d figure only rotate itsxyz-projection without otherwise changing it; all angles and lengths remain the same, interior edges remain interior and exterior remain exterior. (Tip:you can also rotate the figure inxyz-space by clicking-and-dragging with your mouse. To rotate inxyw-space press Shift while dragging.)Starting from that same figure and pressing

yw(actually Ctrl- or right-clicking: remember, this just reverses the direction of rotation) eventually effects the following transformation:

yw-rotation essentially changes thexyzprojectionHere the

xyz-projection has changed essentially: for instance some edges have moved from exterior to interior, and vice-versa.Finally, let's get a sense of what is happening with rotations that involve

w. We first introduce the notion of perspective projection. In the 3d-to-2d case, I call it z-perspective; it's standard and the following diagram illustrates it:

standard "z-perspective" projection of 3-cube in initial orientation

We place a viewpoint

pso that the cube is "in front of"p, i.e., the cube lies in the positivezdirection fromp, and place a projection plane parallel to thexy-plane betweenpand the cube^{1}. Rays are traced frompto each point of the figure, and where that ray crosses the projection plane is that point's projection. As shown in the diagram, the face of the 3-cube closest topin thezdirection projects to a square and the farthest one projects to a smaller square inside it.

3-cube, initial position, z-perspective4-cube, (near) initial position, w-perspectivexz-rotation beginsxw-rotation begins~45° : Red face's trailing edge's closest

approach to, and green face's trailing

edge's farthest retreat from,pRed facet's trailing faces's closest

approach to, and green facet's trailing

face's farthest retreat from,qRed face about to turn inside out Red facet about to do the same 90° : Equipoise! Green face about to turn inside out Green facet ditto ~135° : Red face's leading edge's farthest

retreat from, and green face's leading

edge's closest approach to,pRed facet's leading face's farthest

retreat from, and green facet's leading

face's closest approach to,q180° : Tables turned One can use an analogous (and again, standard) method to project a 4-cube from

xyzw-space toxyz-space, which I callw-perspective. One places a viewpointqso that the 4-cube lies in the positivewdirection from (i.e. "in front of") it, places a 3-dimensionalxyzhyperplanein front ofq, traces rays fromqto the points of the figure, and projects each point to where its ray crosses the hyperplane. Comparing a 3-cube shown inz-perspective and 4-cube shown inw-perspective, both in initial position (jostling the 4-cube slightly so as to be able to see it all), we see that the 4-cube's projection has a (3-)cube-within-a-cube corresponding to the 3-cube's square-within-a-square. The outer-projecting 3-cube facet (shown in red) is the one closest toqin thewdirection, the inner-projecting one (shown in green) is the farthest.Now to the rotation. First understand that with

xw-rotation every point'sxandwcoordinates are changing in exactly the way thatxandychange when you do anxy-rotation, viz. rotate something about thez-axis in 3-space, and the other coordinates don't change at all^{2}. Thewdimension is no different than any of the others, it was just unlucky enough to arrive last. Rotate the 3-cube withxzand the 4-cube withxw. Even though the faces of the 3-cube are all the same fixed size, as the red and green rotate to each other's former positions, the first's 2-d projection (eventually) shrinks and the second's grows, and both exchange their own inside and outside. Not remarkable for us since we innately understand that the actual sizes and angles are fixed and automatically interpret projected size as an indicator ofz-distance. The trick is to see the relationship between the 3-cube's rotation and its 2-d projection, and understand that the same thing is happening with the 4-cube and its 3-d projection: in the full 4-dimensional hypercube all edges are the same length and all angles are right angles, but in the projected 3-space figure, those lengths (and so necessarily the angles) change according to each edge'sw-distance from the viewpointq. A simple rotation in 4-space causes the 3-space projection to turn inside out.You can also find discussion and animations of this phenomenon here, with an artist's viewpoint

andthe math shown explicitly if you want it.

- For tips and tricks with rotations, see here (i.e. RTFM :-)
- Wherever it makes sense, e.g. with rotations and zooming, right- or Ctrl- click the control for reverse action.
- Click
options...to try different rendering options.z-shadingand usingdashorcolorto distinguish between the projected figure's interior and exterior can help you see how thexyzprojection sits in 3-space.- Try marking or labelling different parts of the figure with vertex labelling, edge coloring, and/or d-cell coloring so that you can follow them as the polytope rotates.

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- Actually, since we can extend the rays forever, we can place the plane
anywherein front ofp, not necessarily betweenpand the figure: in practice the plane is usually conceived as being a fixed distance in front ofpaspor the viewed objects move, equivalent to using a camera lens with a fixed focal length. 1- The notion of rotation about an axis, so natural and useful in 3-space, doesn't carry over at all to higher dimensions. (In fact in 4-space there is no such thing. Consider a 4-space "general rotation"
R, by which I mean a transformation where the origin is left fixed, all distances (and hence all angles) are preserved, and "orientation is preserved", i.e. no mirror images - a so-called "special orthogonal transformation". IfRleaves an axis fixed it must leave a whole plane fixed, for the very reason that the only "general rotations" in 3-space are simple rotations about an axis. The difference is that a general rotation in 4-spacemayleave no point but the origin fixed. Examples are easy - for instance, follow anxy-rotation with azw-rotation. The net transformation leaves no point but the origin fixed, since every other point has some non-zero coordinate; one of the two planar rotations will change it and the other will leave it alone.) One could refer to what we call anxy-rotation or a rotation "in thexy-plane" as a rotation "about thezw-plane", but what would be the point? Why not name it with the coordinates where something is actually happening. 2