Cut Depths¶
3D puzzles¶
Tetrahedron¶
The tetrahedron is self-dual, so every vertex-turning tetrahedron has an identical facet-turning tetrahedron (and vice versa).
Facet-turning cut depths¶
Assume each facet is at distance \(1\) from the origin. Then each vertex is at distance \(-3\) along the same vector. Thus each cut is in the range \((-3, 1)\).
| Common Name | Technical Name | Cut distance | Piece types |
|---|---|---|---|
| Halpern-Meier Tetrahedron | 4-Simplex C | \(\left( -\frac{1}{3}, 1 \right)\) | 1g + 2g + 3g |
| Tetraminx | 4-Simplex B | \(-\frac{1}{3}\) | 2g + 3g |
| 4-Simplex A | \(\left( -1, -\frac{1}{3} \right)\) | 2g + 3g + 4g | |
| 2-Layer Pyraminx | \(\left( -3, -1 \right]\) | 3g + 4g |
Piece types¶
| Name | Grips |
|---|---|
| Core | 0 |
| Center | 1 |
| Edge | 2 |
| Corner | 3 |
| Anticore | 4 |
Vertex-turning cut depths¶
Each facet-turning cut distance \(d_v\) is equivalent to the facet-turning cut_depth \(d_f = -d_v\). Thus each each cut is in the range \((-1, 3)\).
| Common Name | Technical Name | Cut distance | Piece types |
|---|---|---|---|
| 2-Layer Pyraminx | \(\left[ 1, 3 \right)\) | 0g + 1g | |
| 4-Simplex A | \(\left( \frac{1}{3}, 1 \right)\) | 0g + 1g + 2g | |
| Tetraminx | 4-Simplex B | \(\frac{1}{3}\) | 1g + 2g |
| Halpern-Meier Tetrahedron | 4-Simplex C | \(\left( -1, \frac{1}{3} \right)\) | 1g + 2g + 3g |
4-Simplex Pyraminx¶
The 3-Layer 4-Simplex Pyraminx is a combination of the 2-Layer 4-Simplex Pyraminx with 4-Simplex A.
Dodecahedron¶
Facet-turning cut depths¶
Assume each facet is at distance \(1\) from the origin. Thus each cut is in the range \([0, 1)\).
Note that \(\phi = \frac{1 + \sqrt 5}{2}\).
| Name | Cut distance | Piece types |
|---|---|---|
| Megaminx | \(\left[ \frac{1}{\phi}, 1 \right)\) | 1g, 2g, 3g |
| Megaminx Crystal | \(\left( \frac{1}{\sqrt 5}, \frac{1}{\phi} \right)\) | 1g, 2g, 3g, 4g |
| Pyraminx Crystal | \(\frac{1}{\sqrt 5}\) | 3g, 4g |
| Litestarminx, Curvy Starminx | \(\left( \frac{2}{\phi} - 1, \frac{1}{\sqrt 5} \right)\) | 3g, 4g, 5g, 6g (PU center) |
| Starminx | \(\frac{2}{\phi} - 1\) | 4g, 5g, 6g (PU center) |
| Master Pentultimate | \(\left( 0, \frac{2}{\phi} - 1 \right)\) | 4g, 5g, 6g (PU center), 6g (PU corner) |
| Pentultimate | \(0\) | 6g (PU center), 6g (PU corner) |
Piece types¶
| Name | Grips |
|---|---|
| Core | 0 |
| Megaminx center | 1 |
| Megaminx edge | 2 |
| Megaminx corner | 3 |
| Crystal edge | 4 (F U R L) |
| Crystal petal | 5 (F U R L DR) |
| Pentultimate center | 6 (F U R L DR DL) |
| Pentultimate corner | 6 (F U R L DR BR) |
Icosahedron¶
Facet-turning cut depths¶
Assume each facet is at distance \(1\) from the origin. Thus each cut is in the range \([0, 1)\).
Note that \(\phi = \dfrac{1 + \sqrt 5}{2}\).
| Name | Cut distance | Piece types |
|---|---|---|
| Radiolarian 1.5 | \(\left( \frac{\sqrt{5}}{3}, 1 \right)\) | 1g, 2g, 3g, 4g (Wing), 5g (Corner) |
| Radiolarian 2 | \(\frac{\sqrt{5}}{3}\) | 2g, 3g, 4g (Wing), 5g (Corner) |
| Radiolarian 3 | \(\left( \frac{1}{\phi}, \frac{\sqrt{5}}{3} \right)\) | 2g, 3g, 4g (Wing), 4g (Center), 5g (Corner) |
| Radiolarian 4 | \(\frac{1}{\phi}\) | 3g, 4g (Wing), 4g (Center), 5g (Corner) |
| Radiolarian 4.5 | \(\left( \frac{4+\sqrt{5}}{11}, \frac{1}{\phi} \right)\) | 3g, 4g (Wing), 4g (Center), 5g (Corner), 5g (Petal), 6g (Edge) |
| Radiolarian 5 | \(\frac{4+\sqrt{5}}{11} = \frac{\phi^2}{2+\phi^2}\) | 4g (Wing), 4g (Center), 5g (Corner), 5g (Petal), 6g (Edge) |
| Radiolarian 6 | \(\left( \frac{4+\sqrt{5}}{11}, \frac{1}{\phi} \right)\) | 4g (Wing), 4g (Center), 5g (Corner), 5g (Petal), 6g (Edge), 6g (Petal) |
| Radiolarian 7 | \(5 - 2\sqrt(5)\) | 4g (Center), 5g (Corner), 5g (Petal), 6g (Edge), 6g (Petal) |
| Radiolarian 8 | \(\left[ \frac{3-\sqrt{5}}{2}, 5 - 2\sqrt{5} \right)\) | 4g (Center), 5g (Corner), 5g (Petal), 6g (Edge), 6g (Petal), 7g |
| Radiolarian 8.5 | \(\left( \frac{1}{3}, \frac{3-\sqrt{5}}{2} \right)\) | 4g (Center), 5g (Corner), 5g (Petal), 6g (Edge), 6g (Petal), 7g, 8g (Edge) |
| Radiolarian 9 | \(\frac{1}{3}\) | 5g (Corner), 6g (Petal), 7g, 8g (Edge) |
| Radiolarian 10 | \(\left( \frac{1}{\phi^3}, \frac{1}{3} \right)\) | 6g (Petal), 7g, 8g (Edge), 8g (Kite), 9g (Petal), 10g (Center) |
| Radiolarian 11 | \(\frac{1}{\phi^3} = \sqrt{5}-2\) | 7g, 8g (Edge), 8g (Kite), 9g (Petal), 10g (Center) |
| Radiolarian 12 | \(\left( 1-\frac{2}{\sqrt{5}}, \frac{1}{\phi^3} \right)\) | 7g, 8g (Edge), 8g (Kite), 9g (Petal), 9g (Wing), 10g (Center), 10g (Corner) |
| Radiolarian 13 | \(1-\frac{2}{\sqrt{5}}\) | 8g (Edge), 8g (Kite), 9g (Petal), 9g (Wing), 10g (Center), 10g (Corner) |
| Radiolarian 14 | \(\left( 0, 1-\frac{2}{\sqrt{5}} \right)\) | 8g (Edge), 8g (Kite), 9g (Petal), 9g (Wing), 10g (Center), 10g (Corner), 10g (Wing) |
| Radiolarian 15 | \(0\) | 10g (Center), 10g (Corner), 10g (Wing) |
Piece types¶
| Name | Grips |
|---|---|
| Core | 0 |
| Stationary Center | 1 |
| Shallow Edge | 2 |
| Shallow Corner | 5 |
| Shallow Petal | 3 |
| Shallow Corner Petal | 4 |
| Shallow Moving Center | 4 |
| Intermediate Edge | 6 |
| Shallow Center Adjacent | 5 |
| Shallow Corner Adjacent | 6 |
| Edge Wing | 7 |
| Deep Edge | 8 |
| Deep Center | 10 |
| Deep Center Adjacent | 9 |
| Kite | 8 |
| Deep Corner | 10 |
| Deep Edge Wing | 9 |
| Deep Corner Petal | 10 |
4D puzzles¶
4-Simplex¶
The 4-simplex is self-dual, so every vertex-turning 4-simplex has an identical facet-turning 4-simplex (and vice versa).
Facet-turning cut depths¶
Assume each facet is at distance \(1\) from the origin. Then each vertex is at distance \(-4\) along the same vector. Thus each cut is in the range \((-4, 1)\).
| Name | Cut distance | Piece types |
|---|---|---|
| 4-Simplex D | \(\left( -\frac{1}{4}, 1 \right)\) | 1g + 2g + 3g + 4g |
| 4-Simplex C | \(-\frac{1}{4}\) | 2g + 3g + 4g |
| 4-Simplex B | \(\left( -\frac{2}{3}, -\frac{1}{4} \right)\) | 2g + 3g + 4g + 5g |
| 4-Simplex A | \(\left( -\frac{3}{2}, -\frac{2}{3} \right]\) | 3g + 4g + 5g |
| 2-Layer 4-Simplex Pyraminx | \(\left( -4, -\frac{3}{2} \right]\) | 4g + 5g |
Piece types¶
| Name | Grips |
|---|---|
| Core | 0 |
| Center | 1 |
| Ridge | 2 |
| Edge | 3 |
| Corner | 4 |
| Anticore | 5 |
Vertex-turning cut depths¶
Each facet-turning cut distance \(d_v\) is equivalent to the facet-turning cut_depth \(d_f = -d_v\). Thus each each cut is in the range \((-1, 4)\).
| Name | Cut distance | Piece types |
|---|---|---|
| 2-Layer 4-Simplex Pyraminx | \(\left[ \frac{3}{2}, 4 \right)\) | 0g + 1g |
| 4-Simplex A | \(\left[ \frac{2}{3}, \frac{3}{2} \right)\) | 0g + 1g + 2g |
| 4-Simplex B | \(\left( \frac{1}{4}, \frac{2}{3} \right)\) | 0g + 1g + 2g + 3g |
| 4-Simplex C | \(\frac{1}{4}\) | 1g + 2g + 3g |
| 4-Simplex D | \(\left( -1, \frac{1}{4} \right)\) | 1g + 2g + 3g + 4g |
4-Simplex Pyraminx¶
The 3-Layer 4-Simplex Pyraminx is a combination of the 2-Layer 4-Simplex Pyraminx with 4-Simplex A.