Add hypercubing standards

README.md

@@ -10,6 +10,7 @@ Contribute [on GitHub](https://github.com/cubing/standards).
 4. [3x3x3 State Representations](./draft/4/3x3x3-state-representations/) (Draft)
 5. [Binary 3x3x3 Encoding](./draft/5/binary-3x3x3-encoding/) (Draft)
 6. [SiGN and LGN](./draft/6/sign-lgn-notation/) (Draft)
+7. [4D Projection and Basic Hypercube Twists](./draft/7/4d-projection-and-basic-hypercube-twists/) (Draft)
 
 ## Future Ideas
 

draft/7/4d-projection-and-basic-hypercube-twists/index.md

@@ -0,0 +1,128 @@
+# Cubing Standard #7: 4D Projection and Basic Hypercube Moves (DRAFT)
+
+Status: `DRAFT`  
+Prescriptive or descriptive: descriptive (this standard attempts to describe existing conventions)
+
+This standard describes the basic moves for a N×N×N×N facet-turning hypercube.
+
+## 7.1 Polytope elements
+
+When possible, pieces are named according to the rank of the element of the polytope they correspond to. Let _d_ be the rank (i.e., dimension) of the puzzle.
+
+| Rank of element | Name   |
+| --------------- | ------ |
+| 0               | Corner |
+| 1               | Edge   |
+| 2               | Face   |
+| 3               | Cell   |
+| _d_ - 3         | Peak   |
+| _d_ - 2         | Ridge  |
+| _d_ - 1         | Facet  |
+
+These terms are taken from [Polytope - Wikipedia](https://en.m.wikipedia.org/wiki/Polytope#Elements), with the exception of "corner" instead of "vertex."
+
+The terms "corner" and "edge" are preferred over "peak," "ridge," and "facet." For puzzles in 4 or more dimensions, the terms "peak", "ridge", and "facet" are preferred over "face" and "cell."
+
+For a 4D hypercube, the terms "corner," "edge," "ridge," and "facet" are preferred, although "cell" is also in common use. In 4D specifically, "facet" and "cell" are interchangeable.
+
+## 7.1 Facets
+
+There are 8 facets of a 4-dimensional hypercube, along with the color in standard orientation:
+
+| Facet letter | Facet name | Vector     | Color  |
+| ------------ | ---------- | ---------- | ------ |
+| `R`          | Right      | positive X | Red    |
+| `L`          | Left       | negative X | Orange |
+| `U`          | Up         | positive Y | White  |
+| `D`          | Down       | negative Y | Yellow |
+| `F`          | Front      | positive Z | Front  |
+| `B`          | Back       | negative Z | Back   |
+| `O`          | Out        | positive W | Pink   |
+| `I`          | In         | negative W | Purple |
+
+Note: historically, `K` ("kata") has been used instead of `O`, and `A` ("ana") or `T` ("top") have been used instead of `I`. It is recommended to avoid these because it is confusing which one corresponds to which cell. Furthermore, `A` conflicts with 5D notation (`A` for "anterior").
+
+These also define the standard projection:
+
+- Positive X pointing to the right
+- Positive Y pointing up
+- Positive Z pointing forward toward the 3D camera
+- Positive W pointing forward toward the 4D camera
+
+The `O` facet is typically hidden, but the other seven facets are typically visible and spaced apart. The default 3D camera may have a rotational offset applied to give a better view of the 7 facets as follows:
+
+- `R` may point more forward and down
+- `F` may point more left and down
+- `U` may point more forward
+
+## 7.2 Clockwise Moves
+
+Moves are named using 2, 3, or 4 letters, each corresponding to a facet. All letters are uppercase, with no space or punctuation between them. The first letter indicates which facet is rotated; the remaining letters describe the polytope element that is fixed by the move. Letters other than the first one may be specified in any order. The move occurs in a 2D plane, over the minimum nonzero angle necessary to restore the puzzle to its original shape.
+
+| Move type   | Number of letters | Angle | Example |
+| ----------- | ----------------- | ----- | ------- |
+| Ridge move  | 2                 | 90°   | `IR`    |
+| Edge move   | 3                 | 180°  | `OUR`   |
+| Corner move | 4                 | 120°  | `DUFR`  |
+
+Rotations appear clockwise when the 4D camera is facing along the vector of the first facet and the 3D camera is facing along the _opposite_ vector from the polytope element described by the second facet. For example, the move `IR` moves a layer of the `I` facet from positive Z to positive Y, which keeps the vector along the X axis fixed.
+
+Examples:
+
+| Move name | Description                                                              |
+| --------- | ------------------------------------------------------------------------ |
+| `IR`      | `R` twists 90° from `F` to `U`                                           |
+| `OUR`     | `O` twists 180°, such that `U` and `R` swap and `F` and `B` swap         |
+| `DUFR`    | `D` twists 120° clockwise, such that `F` goes to `I` and `I` goes to `R` |
+
+In software with mouse controls, these moves are typically executed by right-clicking on the described sticker. For example, `DUFR` is executed by right-clicking on the `D` sticker of the `DUFR` corner piece.
+
+## 7.3 Double Moves
+
+Ridge moves may be doubled by appending `2` to the move. For example, `DF2` denotes the `DF` move repeated twice.
+
+Edge moves have order 3 and so do not need to be doubled.
+
+Corner moves have order 2 and so do not need to be doubled.
+
+## 7.4 Counterclockwise Moves
+
+Moves may be suffixed with `'` to invert them. Alternatively, all but the first letter may be replaced with the letter of the opposite facet. For example, the inverse of `IUR` is `IUR'` or `IDL`.
+
+## 7.5 Layers
+
+Moves may be prefixed with a layer or layer range using SiGN Notation.
+
+## 7.5 Slice Moves
+
+There are four middle slice layers. Slice moves are constructed from facet moves on a parallel facet according to the following table.
+
+| Letter | Mnemonic     | Parallel facet |
+| ------ | ------------ | -------------- |
+| `M`    | "Middle"     | `L`            |
+| `E`    | "Equitorial" | `D`            |
+| `S`    | "Standing"   | `F`            |
+| `P`    | "Planetary"  | `O`            |
+
+For example, `MD` is equivalent to `2LD`
+
+## 7.6 xyz Notation
+
+Ridge moves may instead be specified using a facet letter followed by a lowercase letter `x`, `y`, or `z`. Each combination of a facet and lowercase `x`/`y`/`z` maps to a ridge turn based on its projection in 3D. In the following table, `_` represents the facet letter.
+
+| Facet | `_x` | `_y` | `_z` |
+| ----- | ---- | ---- | ---- |
+| `R`   | `RO` | `RU` | `RF` |
+| `L`   | `LI` | `LU` | `LF` |
+| `U`   | `UR` | `UO` | `UF` |
+| `D`   | `DR` | `DI` | `DF` |
+| `F`   | `FR` | `FU` | `FO` |
+| `B`   | `BR` | `BU` | `BI` |
+| `O`   | `OL` | `OD` | `OB` |
+| `I`   | `IR` | `IU` | `IF` |
+
+For example, `Fz` is equivalent to `FO`.
+
+This notation is generally preferred for puzzle keybinds, for Melinda's 2×2×2×2, and for other physical puzzles. It should be avoided otherwise.
+
+Moves using xyz notation can be grouped using parentheses. For example `OUFR` is equivalent to `(Oz Ox2)`.