I had a look at the 'tiger' on the wiki at

http://teamikaria.com/hddb/wiki/Tiger . It is quite interesting, although little is revealed there.

The equation given is " (sqrt(w^2 + x^2) - a)^2 + (sqrt(y^2 + z^2) -b)^2 = r^2 ". It looks quite frightening. But it disects quite nicely into bi-polar coordinates, where we write (w,x) as (x, \phi), and (y,z) as (y, \psi). The angular coordinates play no part, and we are left with a different equation:

(x-a)^2 + (y-b)^2 = r^2. This is the equation for a circle of radius r, centred on x=a, y=b.

What happens now. The coordinates are valid for x>0, y>0, and there is no dependance on the angular coordinates. So you have this quadrant, and you rotate the y-axis around the wx pland, and you rotate the x axis around the yz plane. Every point on the xy quadrant becomes a 'clifford torus'. A rectangle whose vertices lie at (a,b), (a,0), (0,0), (0,b), gives the duo-cylinder. The line from (a,0) to (0,b) gives the bi-circular tegum. The ellipse quarter from (a,0) to (0,b), gives a bi-circular crind: a kind of equi-ellipsoid, xOoOxOo, which becomes a glome when a=b.

A tiger is then a circle centred at a, b, both greater than r. When the circle is centred at a=0, b or a, b=0, the resulting figure is a spherated circle, or a torus of spherical cross section and circular hole.

Because the tiger is centred on a, b in this scheme, and this is both a "clifford torus", and the margin between the two faces of a duocylinder, we see that there are many descriptions that invoke the duocylinder, simply to effect the necessary clifford torus at the centre of the tiger.

The Clifford TorusIn maths speak, a torus is a rectangle-region, where the sides wrap around like a cylinder at each end. So if you're at the top in column x, the next step upwards is to end at the bottom of the same column x. A particular instance of this is the surface of a doughnut-shape thing in 3d, which is also called a torus. In 4D, the margin between the faces of a duocylinedr is a reasonably undeformed instance of such a rectangle: it shoud be, because it is the prism-product of two circle-surfaces.

The name 'clifford', is usually attached to things that have a relation to 'clifford parallels'. This means that one can divide all of 4-space (or any even number-space), into circles that neither intersect each other, but are none the less concentric on a common centre. For those of you familiar with "complex numbers", it is very easy to verify this, because a 2N space maps onto a product of N argand diagrams, and the multiplication of every point by a value cis(w t), does not change the gradient, and makes every point orbit the centre in one cycle.

The construction of the glome from rotating a xy quadrant over the wx and yz spaces, will sweep the quadrant from 0 to 90 degrees into the climate lattitudes of a 4D planet. 90 is cold, 0 is hot (as in lattitude), and the trace of the point at x degrees (say 27.5), will give all points whose lattitued is 27.5 degrees - a clifford torus.

A clifford torus at a, b gives a rectangle of size 2pi a * 2pi b, the diagonal in either direction, and its parallels (we cut the rectangle from a torus: this can be done anywhere), gives circles, and if the diagonals in all rectangles run the same way, all space is divided into parallel equal-distant circles. Every point on our xy quadrant is mapped onto a set of these: this means any figure that you construct by rotating around the wx and then yz axies, is bounded by a set of concentric circles.

The tigerThe tiger is then a 'spherated bi-circular prism'. I used the term 'glomohedrix' here, explaining that this word meant the surface of a

sphere. Glomohedrix means sphere-surface, but spheres are not used here: circles are.

A tiger = spherated bi-circular prism = spherated bi-

latrix prism.

Spheration turns the point (a,b) into a circle centred thereon, the symmetry by rotations around the wx and yz carry the circle (orthogonally), first into a torus, and then into a bi-torus or tiger. It's just that in the tiger, two of the circles are orthogonal.

Bi-glomohedrix prism are indeed used in relation to clifford-parallels, but not here. The phase space of all of the great circles + directions in 4D, map onto individual points on the 2glmhdx prism. But it's not part of the tiger.

The 'realmic sections' and Cassini ovals, are a bit of a false lead here. People think of toruses in terms of the cross section, not the equation that it might make on the surface of the water as it is being submerged. This is what is implied by being a rototope. This is also what Keiji is hinting at at things like circle # disk etc. The disk is rotated to keep orthogonal to the circle as its orbit sweeps out the torus.

ProductsKeiji is coming to the conclusion that the tiger is ((II)(II)), while the duotorus is (((II)I)I), are different things. In practice, i suspect that it's more due to a mis-understanding on how the products actually work. One suspects by the issues he has with the assorted products what's being implied.

The duotorus and the tiger are both Comb{circle, circle, circle}, in much the same way that the rectangle and square are Prism{line, line}. The tiger is a special instance of a duo-torus, which has bi-cylinderic symmetry.

A prism product is in euclidean geometry, a cartesian product. A Comb product isn't.

Unlike the other products (crind, prism, tegum, pyramid), the comb product, applied to polytopes (rather than euclidean tilings, where it is perfectly regular), is not always applied at the centre. Also, if one fills the resulting surface, one can not suppose that the figures are topologically equal. A surface gives two different topological shapes, not the same. This is why we have sock and hose closures of surfaces.

The reason we have the 'spherated' part is that unless you get down to fixing the order of the comb product, the spherated (which is a kind of thick paint applied to thin things like points and lines), allows us to turn things like circles and bi-circular prisms, into solid things that you can wave about.

The three circles in the tiger, two make the clifford torus, the third runs around the clifford torus, as an outcome of spheration. You can see the spheration circle in the circle we rotate around the torus. Generally, the process of hose and sock places each subsequent circle around the centre of the previous one: so you start with a circle, make it to a 3d torus, and then turn the skin of the 3d torus into a bi-torus.