By the way of holes, a "tiger" might be two "clifford holes" let through the a hollow glome's surface. It really does have two holes, and the holes are reproducable in any number. However, the way one counts holes mathematically is consistant, but gives an answer that is less obvious.
Let's look at holes, and how the tiger is different.
Suppose you start with a model of a cube (in 3d), in which the vertices and edges might be puffed up. How many holes would you think there were in this? Six? You might be supprised to find it has only five holes, although you really can't pick them out. If you fold this down into a
Schlegel diagram, you get a pretzel with five holes in it. You can think of the removal of one face as removing the interior too (ie removing the lid off the box), and the remaining five walls make true holes.
The way you can detect a hole is to pass some kind of sphere-surface that can not disappear. This means that you can change the size without crossing the surface, and some of it will always be seen. The actual nature of the hole is then by putting up a screen to prevent any loops going through the part. The number and type of screens needed to stop any hollow sphere linking is then the "number of holes".
Since a surface divides the space into two, and holes are represented by hollow spheres and their intersecting screens in one space, we can measure the number of holes inside as well as outside. In our model cube-frame, there are loops that can form along any path, that might not disappear. How many breaks do we need to make so that you still have one peice, but no loops: It's five! In three dimensions, the number of holes outside is the same as the number of holes inside. The number of holes is called the 'genus' of the figure.
Four dimensions is different, even for simple holes.
The complement of a hole needing a 2d screen, is one needing 3d. For example, in the spherinder bent to a loop, you have a 2d screen outside, and a 3d screen inside (the join), which when inserted, will stop non-vanishing hollow-spheres.
The 'tiger' has a different kind of hole, which the simple-hole notion can not describe. This hole is probably best described as a puffed-up circle, that is concentric with the glome, and cuts into the surface of the hollow glome. These are (for reasons that shall become apparent soon), "clifford holes". The tiger has two of them.
If you take a duo-cylinder, which we agree is the convex prototype of the tiger, you can turn it into something different by doing this. ( 1 ), replace the two faces by glass windows. (2), replace the margin (clifford torus), by a tiger. (3) drain the interior. You now have something that is a tiger + windows, topologically equal to a hollow glome.
If you just remove one of the windows, you will see that you can pass a 2d plane through it, and a 1d stick, and both of these can be used to 'lift' the tiger. In short, the hole we have created has two different kinds of closure. You can prevent the 2d planes by inserting a 2d screen across the torus. This means, "perpendicular to the height of the prism". In essence, this turns the doughnut-shaped face of the duocylinder back into a bent cylinder, and the 2d plane equates to a non-vanishing 2d circle inside this space.
To stop one poking sticks through it (which corresponds to "points of entry"), you have to fill the thing right up. (i think).
Each hole acts separately, so it has two holes.
Clifford ParallelsWe call these holes "clifford holes", because the hole follows a 'straight line' in spheric space (ie a great circle on a glome), and puffing them up, makes them into the same shape as the holes in the tiger (ie what you get if you bend a long cylinder into a face of a duocylinder).
Straight lines in spheric space (glomochorix = E3), can run equidistant to each other, but are not coplanar. You can for a given great circle, draw an equidistant great circle through any given point.
For those who understand maths, here are some equations to ponder over.
The equation for a straight line is "y = ax + b". When we want it to pass through the origin, we put "y = ax", or "a = y/x". The actual space representing all kinds of 'a' is a semicircle, from y=1, to x=1, to y=-1.
The same holds true for complex numbers, where x, y, a, and b are all complex. We see that x => x+iX, and y => y+iY, maps without distortion onto a four-point x, X, y, Y. So the complex line becomes a 2-space (argand diagram), and the complex plane (x,y), becomes 4-space.
A line is still defined by three points: one at (0,0), one at (x,y), and one at (cx, cy), where 'c' is any complex number. If we write c = r cis(wt), where w = speed of angle, and t = time. We put then that over t fron 0 to 2pi, will make every point circle (0,0), especially since pi itself represents a simple change of sign. This is 'clifford rotation'.
The space of 'clifford rotations' (ie the half-circle), now comes to a sphere whose diameter is x=0, y=0, and x=0, y=1, and y real. This is a 3-sphere, and every point comes to be a non-intersecting clifford-circle.