# Research and mathematical foundations

Research date: 14 September 2026. Scope: the publicly documented mechanics necessary to implement an original 4D exploration game. This is targeted engineering research, not an exhaustive survey of every publication about four-dimensional space. Miegakure's engine and full unreleased campaign are not publicly available to inspect.

## What Miegakure publicly establishes

The [official Miegakure page and FAQ](https://miegakure.com/) describe four spatial coordinates, a visible 3D cross-section, rotation of that section, and later access to translation. Rotating is introduced before direct translation because the latter can move into unseen obstacles. The fourth dimension is explicitly distinct from time. The game uses the consequences of extra spatial freedom as puzzles. These are the reference principles for Aetherfold's staged controls and sealed-room puzzle; Aetherfold's world, progression and implementation are original.

Marc ten Bosch's [November 2020 art-direction article](https://marctenbosch.com/news/2020/11/) explains the coexistence of procedural four-dimensional geometry and conventional meshes embedded with thickness. It also explains why fine detail can flicker during slicing and why broad, readable forms matter. Aetherfold uses broad architectural volumes, continuous sectioning and a conventional character mesh. The final visual revision adds scanned PBR textures, detailed foliage and a more natural camera. The avatar and scanned foliage remain visual props rather than general 4D meshes; gate activation uses all four coordinates.

## Papers consulted

1. Marc ten Bosch, **N-Dimensional Rigid Body Dynamics**, ACM Transactions on Graphics 39(4), 2020. [Author's page](https://marctenbosch.com/ndphysics/) · [Full paper](https://marctenbosch.com/ndphysics/NDrigidbody.pdf) · DOI 10.1145/3386569.3392483. The paper describes a dimension-independent formulation using geometric algebra and generalized collision handling, with a four-dimensional implementation displayed as 3D sections. It establishes that a real higher-dimensional physical simulation is possible. Aetherfold intentionally uses a much smaller custom kinematic controller and static 4D solids; it does not implement this paper's rigid-body/contact solver or claim to.
2. Akira Kageyama, **Keyboard Based Control of Four Dimensional Rotations**, 2016. [Full text and equations](https://arxiv.org/html/1604.02013v1). The paper treats simple rotations in coordinate planes and double rotations in orthogonal plane pairs, with direct keyboard controls. It provides the six-plane basis and rotation-matrix reference for the Observatory. Aetherfold implements elementary plane rotations and their compositions; simultaneous keys can compose rotations, but there is no separate fixed-rate double-rotation preset.
3. Akira Kageyama, **A Visualization Method of Four Dimensional Polytopes by Oval Display of Parallel Hyperplane Slices**, 2016. [Full text](https://arxiv.org/html/1607.01102v1). The paper uses parallel 3D sections to communicate a 4D polytope's structure. It supports the distinction between a section and a projection. Aetherfold renders a single navigable section with a signed-depth compass, not the paper's oval arrangement of multiple views.

The implementation below is a direct derivation for axis-aligned hyperrectangles. It does not depend on copying a paper's source code.

## Coordinates and frame

A physical position is p = (x, y, z, w). Let B = [b₀ b₁ b₂ b₃] be an orthonormal 4×4 matrix whose columns are the slice frame's world-space axes. Let o be the slice origin. A point's local coordinates are:

    q = Bᵀ (p − o)

The visible space is q.w = 0. Its three displayed coordinates are (q.x, q.y, q.z), passed through an ordinary 3D camera. This final 3D-to-screen projection is conventional; the 4D-to-3D step is an intersection, not a perspective projection.

In the campaign, the origin tracks the player's X/Z/W coordinates while its world Y is zero. Rotation preserves the player's actual four-dimensional position. The local frame turns around that player, so the player remains in the slice. An ordinary camera orbit changes neither B nor the four-dimensional position.

## Rotations

For a local coordinate-plane turn of angle θ, replace two columns simultaneously:

    bᵢ′ = cos(θ) bᵢ + sin(θ) bⱼ
    bⱼ′ = −sin(θ) bᵢ + cos(θ) bⱼ

All other columns remain unchanged. The result preserves dot products and distances. At θ = π/2, positive local X movement becomes positive world W movement for an initially aligned XW rotation.

The six coordinate planes are XY, XZ, XW, YZ, YW and ZW. Rotations sharing an axis generally do not commute. Thus the compass uses the current frame rather than pretending that two displayed Euler angles uniquely describe every reachable orientation.

## Hyperrectangle slicing

Each physical box is specified by a four-coordinate center and four positive half-extents. It has 16 vertices, 32 edges and eight cubical boundary cells.

1. Transform its 16 vertices into the current slice frame.
2. For every edge whose endpoints have opposite signs of q.w, compute the crossing parameter t = a.w / (a.w − b.w).
3. Interpolate its full four-coordinate endpoints, retaining the three visible coordinates at that intersection.
4. For each cubical boundary cell, collect and deduplicate the intersected edge points belonging to that cell.
5. Project the cell's outward four-dimensional normal into the slice, order its polygon vertices around that normal and triangulate a fan.
6. If the section misses the volume, the mesh is empty.

This handles arbitrary compositions of all six frame rotations. Degenerate tangent intersections with no three-dimensional volume can collapse to no rendered solid. Floating-point tolerances avoid duplicate vertices at grid-aligned cuts.

A central slice of a unit-half-extent tesseract is a cube of width 2. After a 45° XW turn, its extent along local X is 2√2. Those analytical facts are included as executable checks. Oblique vertices are also transformed back into world space and checked against the original four-dimensional bounds.

## Hyperspheres

For a hypersphere of radius r with signed normal distance d from the slice, the visible sphere has radius:

    r_section = √(r² − d²), when |d| < r

There is no intersection outside that interval. This formula is implemented and analytically tested, and controls the weave-light visibility envelope. The realism revision replaces the initial hypersphere tree canopies with conventional scanned 3D trees. Their anchor positions use four coordinates and they reveal/fade across a finite W thickness. These decorative meshes are an artistic approximation, not exact slices of arbitrary four-dimensional botanical solids. The character and orbiting halo art are also conventional display meshes. The gate’s masonry is now actual rotated 4D box geometry.

## Movement and collision

Ground input moves along the current slice's horizontal basis. Direct fourth-direction input adds a component along b₃. Collision tests use the original world-space four-dimensional solids, rather than the disposable visible meshes. Translation is subdivided to avoid tunneling through narrow walls. The character is approximated by horizontal extents and a vertical body interval. Vertical gravity, jumping, ground contact and head contact are handled separately.

Campaign turns preserve world Y. General vertical-involving rotations are offered in collision-free Observatory flight because a full rotated-gravity character/contact solver is outside this game's implementation. This is an explicit design constraint; it does not turn W into time or reduce the campaign to switched 3D scenes.

## Puzzle validity and boundaries

The cloister has a continuous wall enclosure in X/Z at W = 0. A route search with W fixed cannot reach the interior light. Allowing W movement finds a path around the wall's finite fourth-dimensional extent. The runtime test then follows actual collision-constrained movement and collects the light.

Each realm has a finite exploration boundary. Realm changes are ordinary progression transitions; rotation and W translation inside a realm do not load a replacement scene. Static boxes, decorative-prop anchors, pickups and the player retain their four-dimensional coordinates throughout the transition.

Not implemented: general 4D rigid bodies, moving 4D enemies, arbitrary manifold/CSG geometry, rotational object manipulation, multiplayer, combat, controller support, or a commercial-scale content campaign. The shipped experience is a finite exploration-and-collection puzzle game with a complete beginning and ending.

## Length comparison

Marc ten Bosch reported about 130 puzzles in April 2015 (https://marctenbosch.com/news/2015/04/) and about six main mechanics beyond jumping, view rotation and pushing in February 2018 (https://marctenbosch.com/news/2018/02/the-state-of-the-miegakure-feb-2018/). These are historical development statements, not a measured current playtime. Aetherfold’s 136 chambers include repeated template families and do not establish equivalence to that design depth. No human completion-time study has been performed.