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An extension of the 3x9x15 soap bar.
After building a crazy 44x55x55 barrel, Ilya Toporgilka remembered the balloid/ellipsoid shape in all 3 dimensions and found several ovals with this shape:
5x6
5x7
5x8
6x7
6x8
Ilya Toporgilka thought: "What if the perimeter of the oval with the diameters of 6 and 8 were 24 single segments?" He wanted to see if that gets a larger puzzle in the end than the 3x9x15 soap bar. He took a paper and started the calculations.
Firstly Ilya Toporgilka was searching for the derivative(which puzzle will deform into the given shape of the ellipsoid). That puzzle was a «webbed» 6x6x8 barrel. The next proportional 6n x 6n x 8n cuboids would not outstand the barreling a fortiori.
After barreling into the webbed barrel shape everything goes into the mechanism and it makes curvy cuts, tiny pieces and exposes the sets of the hidden edges. To delete these cuts the exterior parts have to be bandaged to the 2x2x8 and the hidden edges have to be too into single pieces. This further bandages the 8 1x1x4 blocks into single pieces. But the end puzzle is not exactly a 2x2x2. The inventor mentioned that 8 pieces will be «ripped off» and 12 circular edges will show up. Then Ilya Toporgilka understood that the puzzle was an overlapping 2x2x2. By extending this puzzle to infinity we get a 2x2x2 either with the exposed hidden edges getting slimmer and slimmer or a 2x2x2 with coloured internal edges. This puzzle has to be transformed into the ellipsoid which has a perimeter of an oval longer than the «ordinary» but still not looking like a squircle. After thinking Ilya Toporgilka found the way: He took a 60mm 4x4x4, bandaged it into the 2x2x2 and extended it into a 60x60x80mm 2x2x2. He took this brick, cut approx. 13mm from every edge and then in the shape he got he rounded every sharp edge and corner. As well as stickering the outer side, Ilya Toporgilka put stickers in the inside of the puzzle. He means he stickered 12 of the hidden 2x2x2 edges so that they had coloured stickers. The puzzle so that gets the same solving as the fused cube and becomes identical to that.
A 2x2x2 has an internal mechanism of the 3x3x3 with the same 2x2x2 block for alignment purposes. With the internal edges being active and stickered with colours, the puzzle starts acting like the fused cube.
Ilya Toporgilka chose the size 60x60x80mm and 6x6 squares because he recognised at bigger scales the shape cannot be «precisely» carved. If the perfect ellipsoid were made instead, the effect would be gone.
The puzzle contains:
1)Roadblock
Ilya Toporgilka noticed that on the cubic 3x3x7-II puzzle the middle 3 layers not only can mix with each other but also can mix between the other layers at the top and the bottom. The 3x3x9 with the horizontal windows cannot perform that though the 3x3x9 with the vertical windows assembled from 3x3x7 parts can. Ilya Toporgilka noticed that each block is a 1x1x3 cuboid and all of those blocks are connected together forming fully functional siamese puzzles. What Ilya Toporgilka did not like is that each of the blocks is an unproportional 1x1x3.
2)Fully functional siamese puzzles with the universal output.
Ilya Toporgilka classified several outputs of the fully functional siamese puzzles:
a)Vertical output
The exchange between the puzzles is carried out as follows - new pieces are supplied from the puzzle located above into the connecting block thereby pushing the parts of the neighbouring cube.
On Ilya Toporgilka’s puzzle, the pieces are no longer supplied into the connecting block but directly into the puzzle’s elbow. Thanks to the special mechanism, a whirlpool is created which mixes the entire contents of the pieces.
b)Skewed output
The puzzle attaches «into» the puzzle. The example can be a 4x4x4, a 3x3x3 or any combination of 3 natural numbers between 1 and 5 inside a 5x5x5. Thereby each time a cuber solves a professor cube, he also solves these siamese puzzles in addition to that.
c)Horizontal output
The puzzle attaches directly to the other puzzle. The example can be a 3x3x6 cuboid consisting out of 2 fully functional siamese 3x3x3s. They are connected with the horizontal output.
d)Universal output
The puzzle attaches using the parts of the fully functional puzzle, into or directly to the other puzzle. The example can be a 5x5x5 cube consisting from several fully functional siamese 1x1x2s, 1x1x3s, 1x1x4s, 1x1x5s, 1x2x2s, 1x2x3s, 1x2x4s, 1x2x5s, 1x3x3s, 1x3x4s, 1x3x5s, 1x4x4s, 1x4x5s, 1x5x5s, 2x2x2s, 2x2x3s, 2x2x4s, 2x2x5s, 2x3x3s, 2x3x4s, 2x3x5s, 2x4x4s, 2x4x5s, 2x5x5s, 3x3x5s, 3x4x5s, 4x4x5s and 4x5x5s. Those puzzles «inside a puzzle» are not single fully functional puzzles but they form a group of fully functional siamese puzzles with the universal output. The same thing was happening with the fully functional siamese puzzles with a vertical output. Some layers of the puzzles started turning/behaving differently.
The difference between the skewed and horizontal outputs is that, for example, the first 5x5x5 layer(1x5x5 cuboid) stays connected to the opposite 1x5x5 layer using parts from the remaining middle 3x5x5 layers.
On Ilya Toporgilka’s puzzle some of the puzzles inside the puzzle are «button shaped cubes/cuboids».
3)Fortress
There were a few puzzles made with this concept. As an example, a 5x5x5 with 8 mini 2x2x2s(each attached to each corner). However, there was no mix between these puzzles, each one could be scrambled and solved separately. On Ilya Toporgilka’s puzzle the puzzle is attached to every cubie and parts mix between the puzzles.
4)Hypericosagonal barrel
A hyperoctagonal barrel is a siamese 3x3x3 with some extensions added and turned into an oval disc. Ilya Toporgilka’s version is «bigger» version of it. What is more, it allows parts to be mixed between the puzzles. The inventor mentioned the «advantage» that this will not require «so many» puzzles to build and fuse into one. Building the fully functional siamese cubes/cuboids with the vertical output will be impossible or at least very difficult, especially without magnets. There is an issue with the overhanging pieces which would simply fall out(like on the cubes from 7 and upper). So far the solution has been invented only to the Oskar’s fully functional siamese 2x2x2s with the pins being in use.
5)Biemperor icosioctagonal disc
6)Biemperor icosioctagonal prolate spheroid
7)Biemperor icosioctagonal tri-axial ellipsoid
8)Professor 56-gonal disc
9)Professor 56-gonal prolate spheroid
10)Professor 55-gonal tri-axial ellipsoid
11)Master 84-gonal disc
12)Master 84-gonal prolate spheroid
13)Master 84-gonal tri-axial ellipsoid
14)168-gonal disc
15)168-gonal prolate spheroid
16)168-gonal tri-axial ellipsoid
17)Evil twin 36x36x48
There is a bandaged 2x2x2 block which makes the puzzle identical(and with solving) to the fused cube/evil twin.
18)Evil twin 24x36x48
19)Wormhole
There is a puzzle inside the puzzle which behaves differently than the outside puzzle. These pieces have to be «synchronised» with the outer ones. With Ilya Toporgilka’s puzzle it has been achieved by hiding the edges of the fused cube at the inside so that the outside puzzls looks like «a normal» 2x2x2. Unlike Wormholes I, II and III, the puzzle has proportional pieces.
20)Minion.
Ilya Toporgilka had several ideas how to sticker the opposite side. At the end he figured the given shape resembles a Minion from the Despicable me series if turned 90 degrees around one of its axes. Ilya Toporgilka has always wanted to build a minion twisty puzzle since 2014.
21)Cubillusion
The puzzle looks like a prolate spheroid as well as the disc and the tri-axial ellipsoid. The shapes are similar and close to each other. By looking at the puzzle an optical illusion happens how they are changing one another. On Tony Fisher’s Cubillusion one of the cubes was a stickering pattern at the extensions and was not a functional Rubik’s cube. On Ilya Toporgilka’s puzzle all 3 shapes are functional as puzzles. There is also an illusion of:
an elliptical cylinder inside a torus inside a torus
6, 18 and 4 cuboids and several button-shaped puzzles inside a button shaped puzzle
several button shaped puzzles inside a torus inside a torus
22)Fully functional siamese cubillusion
A cubillusion made from fully functional siamese puzzles, some of them are button-shaped.
23)Proportional balloids/ellipsoids a x b x c-II
Initially it seems that those puzzles cannot be made with proportional pieces and only simulated by bandaging a cube. When Ilya Toporgilka had finished making the Hyperball though, he realised that on the round shaped puzzles it is not always clear how to count the number of layers and call the puzzle. They do not have «sides» and «edges».
As en example, a 4x4x4 cube could be a 2x2x16-II puzzle. The second puzzle would have unproportional pieces though. However, if the shape is non-cuboidal and rounded, one can say that this has been achieved.
24)Half proportional(1/2) prolate 18x18x46 spheroid
These series of puzzle are done by bandaging a larger fully functional cuboid.
25)Half proportional(1/3) prolate 12x12x44 spheroid
26)H-nal(1/4) prolate 9x9x42 spheroid
27)H-nal(1/6) prolate 6x6x38 spheroid
28)H-nal(1/9) prolate 4x4x32 spheroid
29)H-nal(1/12) prolate 3x3x26 spheroid
30)H-nal(1/18) prolate 2x2x14 spheroid
31)H-nal(1/2) tri-axial 12x18x46 ellipsoid
32)H-nal(1/3) tri-axial 8x12x44 ellipsoid
33)H-nal(1/4) tri-axial 6x9x42 ellipsoid
34)H-nal(1/6) tri-axial 4x6x38 ellipsoid
35)H-nal(1/12) tri-axial 2x3x26 ellipsoid
36)H-nal(1/2) 12x18x46 disc
37)H-nal(1/3) 8x12x44 disc
38)H-nal(1/4) 6x9x42 disc
39)H-nal(1/6) 4x6x38 disc
40)H-nal(1/12) 2x3x26 disc
41)N or infinite siamese discs,prolate spheroids and tri-axial ellipsoids(combinations of them),where the shape of the following is different to the previous, with a vertical output.
42)Half a (1/2, 1/3, 1/4, 1/5…) of the puzzle No41.
43)Fully functional siamese siamese cubillusion
44-86)Mini 1-43 puzzles,each cubie is 1,(6)7mm
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Hyperball
