Polyhedron Dude wrote:Marek14 wrote:Polyhedron Dude: your renderings are great, but there's not many more 4D toratopes, sadly. Would you consider trying for 5D toratopes as well, in a similar vein to your polyteron images?
If I read my old posts here correctly, there's 12 of them
Some of them, however, would have renderings looking like two spheres or torii inside of each other -- those might be better rendered with, say, one octant missing so the inner structure could be seen...
I plan on doing those
Just building a quick list (also to get used to new names)... Might contain errors. In particular, I'm not sure how exactly to describe two displaced ditoruses. ((((I)I)I)I), (((II)(I))I) and (((II)I)(I)) are all different positions, first is encountered in cuts of tritorus, second in tigric torus and third in cyltorintigroid.
Pentasphere (IIIII): Origin is a sphere. Axes are glomes, so the spheres will shrink along them. Graph has circular symmetry.
Toratesserinder ((II)III): three possible coordinate renderings, if displayed as ((xy)zwv), they are xy, xz and zw.
xy: Origin is empty. Axes are pairs of displaced glomes, so each will have a sphere appear, grow, shrink and disappear in both directions. Graph has circular symmetry.
xz: Origin is a pair of displaced spheres. Axes are pairs of displaced glomes (x) and toracubinder (z). In x direction, the spheres will shrink until they disappear. In z direction, spheres will transform into rotated Cassini ovals, merge, then disappear.
zw: Origin is a torus. Axes are toracubinders. In both directions, the inner diameter of torus will shrink until it reduces to a circle and disappears. Graph has circular symmetry.
Toraduocyldyinder ((II)(II)I): three possible coordinate renderings, if displayed as ((xy)(zw)v), they are xy, xz and xv.
xy: Origin is empty. Axes are pairs of vertically stacked toracubinders. In both directions, a circle will appear, then fatten into a torus and back into a circle. Graph has circular symmetry.
xz: Origin is four spheres in vertices of a rectangle. Axes are pairs of vertically stacked toracubinders. In both directions, two pairs of spheres will merge, then disappear; however, in one direction the horizontal pairs will merge, in the other direction the vertical pairs.
xv: Origin is two toruses, vertically stacked. Axes are a pair of vertically stacked toracubinders (x) and a tiger (v). In x direction, inner diameter of both toruses will shrink until they reduce to circles and disappear. In v direction, toruses will merge and eventually reduce to a circle and disappear.
Toracubspherinder ((III)II): three possible coordinate renderings, if displayed as ((xyz)wv), they are xy, xw and wv.
xy: Origin is a pair of displaced spheres. Axes are toracubinders. In both directions, spheres will merge and disappear. Graph has circular symmetry.
xw: Origin is a torus. Axes are a toracubinder (x) and a toraspherinder (w). In x direction, inner diameter of torus will shrink until it reduces to a circle and disappears. In w direction, the hole will become filled and the torus eventually shrinks and disappears.
wv: Origin is a pair of concentric spheres. Axes are toraspherinders. In both directions, spheres will come closer until they merge and disappear. Graph has circular symmetry. Images should be cut open.
Toracubtorinder (((II)I)II): five possible coordinate renderings, if displayed as (((xy)z)wv), they are xy, xz, xw, zw and wv.
xy: Origin is empty. Both axes are pairs of toracubinders next to each other. In both directions, points will appear, separate into two spheres, then re-merge and disappear. Graph has circular symmetry.
xz: Origin is four spheres arranged in a line. Axes are a pair of toracubinders next to each other (x) and a pair of toracubinders differing in their outer diameters (z). In x direction, each pair of spheres will merge and disappear. In z direction, inner pair of spheres will merge and disappear, after that the two remaining spheres will merge and disappear.
xw: Origin is a pair of toruses next to each other. Axes are a pair of toracubinders next to each other (x) and a ditorus (w). In x direction, inner diameters of both toruses will shrink until they reduce to circles and disappear. In w direction, the two toruses will merge and eventually disappear.
zw: Origin is a pair of toruses differing in their outer diameters. Axes are a pair of toracubinders differing in their outer diameters (z) and a ditorus (w). In z direction, inner diameters of both toruses will shrink until they reduce to circles and disappear. In w directions, the two toruses will merge into one and eventually reduce to a circle and disappear.
wv: Origin is a pair of two cocircular toruses. Both axes are ditoruses. In both directions, the toruses will come closer until they merge and disappear. Graph has circular symmetry. Images should be cut open.
Cylspherintigroid ((III)(II)): three possible coordinate renderings, if displayed as ((xyz)(wv)), they are xy, xw and wv.
xy: Origin is a pair of vertically stacked toruses. Both axes are tigers. In both directions, the toruses will merge until they become one and eventually reduce to a circle and disappear. Graph has circular symmetry.
xw: Origin is a pair of vertically stacked toruses. Axes are a tiger (x) and a pair of vertically stacked toraspherinders (w). In x direction, the toruses will merge until they become one and eventually reduce to a circle and disappear. In w direction, holes in toruses become filled and they eventually disappear.
wv: Origin is empty. Both axes are pairs of vertically stacked toraspherinders. In both directions, a sphere appears, splits in two concentric spheres, then merges again and disappears. Graph has circular symmetry. Image should be cut open.
Cyltorintigroid (((II)I)(II)): five possible coordinate renderings, if displayed as (((xy)z)(wv)), they are xy, xz, xw, zw and wv.
xy: Origin is empty. Both axes are pairs of tigers next to each other. In both directions, a circle appears, fattens into torus, splits in a pair of vertical toruses, then merges, reduces and disappears. Graph has circular symmetry.
xz: Origin is four vertically stacked toruses. Axes are pairs of tigers next to each other (x) and two tigers differing in one of their outer diameters (z). In x direction, outer pairs of toruses merge, then reduce to a circle and disappear. In z direction, inner pair of toruses merges, reduces to a circle, then disappears, then the remaining pair does the same.
xw: Origin is two pairs of vertically stacked toruses next to each other. Axes are a pair of tigers next to each other (x) and a type 3 pair of ditoruses w). In x direction, each vertical pair of toruses merges, reduces to a circle and disappears. In w direction, each horizontal pair of toruses merges and eventually disappears.
zw: Origin is two pairs of vertically stacked toruses differing in their outer diameters. Axes are two tigers differing in one of their outer diameters (z) and a type 3 pair of ditoruses (w). In z direction, each vertical pair of toruses merges and eventually reduces to a circle and disappears. In w direction, each concentric pair of toruses merges and eventually reduces to a circle and disappears.
wv: Origin is empty. Both axes are type 3 pairs of ditoruses. In both directions, circle appears, becomes a torus, splits in a pair of toruses differing in their outer diameters, then re-merges, reduces and disappears. Graph has circular symmetry.
Toraglominder ((IIII)I): two possible coordinate renderings, if displayed as ((xyzw)v), they are xy and xv.
xy: Origin is a torus. Both axes are toraspherinders. In both directions, the hole becomes filled and the torus reduces to a point and disappears. Graph has circular symmetry.
xv: Origin is a pair of concentric spheres. Axes are a toraspherinder (x) and a pair of concentric glomes (v). In x direction, the spheres come together, merge and disappear. In v direction, the spheres shrink until the inner, and then the outer sphere disappears. Image should be cut open.
Cylindrical ditorus (((II)II)I): five possible coordinate renderings, if displayed as (((xy)zw)v), they are xy, xz, xv, zw and zv.
xy: Origin is empty. Both axes are pairs of toraspherinders next to each other. In both directions, a point appears, grows, gets a hole to become a torus, then fills it, shrinks and disappears. Graph has circular symmetry.
xz: Origin is a pair of toruses next to each other. Axes are a pair of toraspherinders next to each other (x) and a ditorus (z). In x direction, each torus fills its hole and disappears. In z direction, the toruses merge and eventually disappear.
xv: Origin is two displaced pairs of concentric spheres. Axes are a pair of toraspherinders next to each other (x) and a pair of cocircular toracubinders (v). In x direction, each pair of concentric spheres comes closer, merges and disappears. In v direction, outer, then inner spheres merge, shrink and inner, then outer spheres disappear. Image should be cut open.
zw: Origin is a pair of toruses differing in their outer diameters. Both axes are ditoruses. In both directions, the toruses merge, then reduce to a circle and disappear. Graph has circular symmetry.
zv: Origin is a pair of cocircular toruses. Axes are a ditorus (z) and a pair of cocircular toracubinders (v). In z direction, toruses come closer, merge and disappear. In v direction, each torus shrinks its inner diameter, reduces to circle, then disappears. Image should be cut open.
Tigric torus (((II)(II))I): three possible coordinate renderings, if displayed as (((xy)(zw))v), they are xy, xz and xv.
xy: Origin is empty. Both axes are type 2 pairs of ditoruses. In both directions, torus appears, splits in two cocircular toruses, then re-merges and disappears. Graph has circular symmetry. Image should be cut open.
xz: Origin is four toruses arranged in vertices of a rectangle. Both axes are type 2 pairs of ditoruses. In both directions, two pairs of toruses merge and eventually disappear, but in one of them vertical pairs merge, in the other the horizontal pairs.
xv: Origin is two vertically stacked pairs of cocircular toruses. Axes are a type 2 pair of ditoruses (x) and two comarginal tigers (v). In x direction, the cocircular pairs come closer, merge and disappear. In v direction, the vertical pairs merge (outer, then inner), reduce to a circle (inner, then outer) and disappear. Image should be cut open.
Spheric ditorus (((III)I)I): four possible coordinate renderings, if displayed as (((xyz)w)v), they are xy, xw, xv and wv.
xy: Origin is a pair of toruses next to each other. Both axes are ditoruses. In both directions, the toruses merge and eventually disappear. Graph has circular symmtery.
xw: Origin is a pair of toruses differing in their outer diameters. Axes are a ditorus (x) and a pair of toraspherinders differing in their outer diameters (w). In x direction, toruses merge, reduce to a circle and disappear. In w direction, inner torus fills its hole, shrinks and disappears, then the outer torus fills its hole, shrinks and disappears.
xv: Origin is a pair of cocircular toruses. Axes are a ditorus (x) and a pair of cospherical toraspherinders (v). In x direction, toruses come closer, merge and disappear. In v direction, toruses fill their holes (outer, then inner), shrink and disappear (inner, then outer). Image should be cut open.
wv: Origin is four concentric spheres. Axes are a pair of toraspherinders differing in their outer diameters (w) and a pair of cospherical toraspherinders (v). In w direction, two outer pairs of spheres come closer, merge and disappear. In v direction, two inner spheres merge, then disappear, then two outer spheres merge, then disappear. Image should be cut open.
Tritorus ((((II)I)I)I): seven possible coordinate renderings, if displayed as ((((xy)z)w)v), they are xy, xz, xw, xv, zw, zv and wv.
xy: Origin is empty. Both axes are type 1 pairs of ditoruses. In both directions, complete cut of ditorus with two toruses in the middle appears. Graph has circular symmetry.
xz: Origin is four toruses lying in a line. Axes are a type 1 pair of ditoruses (x) and a pair of ditoruses differing in their outer diameter (z). In x direction, two outer pairs of toruses merge and eventually disappear. In z direction, inner pair of toruses merges first, then the outer pair.
xw: Origin is two pairs of toruses differing in their outer diameters lying next to each other. Axes are a type 1 pair of ditoruses (x) and a pair of ditoruses differing in their middle diameter (w). In x direction, each concentric pair merges, reduces to a circle and disappears. In w direction, the outer toruses merge, then the inner, eventually inner disappears, then outer.
xv: Origin is two pairs of cocircular toruses lying next to each other. Axes are a type 1 pair of ditoruses (x) and a pair of cotoroidal ditoruses (v). In x direction, each pair of toruses comes closer, merge, then disappear. In v direction, the outer toruses merge, then the inner, eventually inner disappears, then outer. Image should be cut open.
zw: Origin is four concentric toruses differing in their outer diameters. Axes are a pair of ditoruses differing in their outer diameter (z) and a pair of ditoruses differing in their middle diameter (w). In z direction, each outer pair of toruses merges, reduces to a circle, then disappears. In w direction, inner pair of toruses merges, then disappears, then the outer pair.
zv: Origin is two concentric pairs of cocircular toruses differing in their outer diameters. Axes are a pair of ditoruses differing in their outer diameter (z) and a pair of cotoroidal ditoruses (v). In z direction, each pair of toruses merges, then disappears. In v direction, outer toruses merge, then inner toruses, then inner toruses reduce to a circle and disappear, then outer toruses. Image should be cut open.
wv: Origin is four cocircular toruses. Axes are a pair of ditoruses differing in their middle diameter (w) and a pair of cotoroidal ditoruses (v). In w direction, outer pairs of toruses merge, then disappear. In v direction, inner pair of toruses merges and disappears, then the outer pair. Image shoud be cut open.
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