The Tesseract
A tesseract is the four-dimensional analogue of a cube. Just as a cube has 8 corners, 12 edges, 6 faces and 1 volume, a tesseract has 16 corners, 32 edges, 24 faces, 8 cubic cells and 1 four-dimensional hypervolume.
In Hypercube 4D, the tesseract is used as a game board with a side length of 4 in every dimension:
4 × 4 × 4 × 4 = 256 cells
The key insight: a tesseract looks exactly like a 4×4 grid of 4×4 grids. This is not an approximation — it is a mathematically exact projection of the 4D structure onto 2D.
The Coordinate System
Every cell on the board is uniquely identified by four coordinates written as a vector:
(x, y, z, w)
Each coordinate ranges from 1 to 4. The axes are split into two levels:
Local axes — inside a sector
x is the column within a 4×4 sector (1 = left, 4 = right).
y is the row within a 4×4 sector (1 = top, 4 = bottom).
Global axes — sector position
z is the column of the sector in the 4×4 macro grid.
w is the row of the sector in the 4×4 macro grid.
Example: the coordinate (1, 2, 3, 1) means: column 1, row 2 inside the sector at macro-column 3, macro-row 1.
The Macro & Micro Trick
The most important mental model for 4D navigation is the macro/micro equivalence:
The game board behaves identically at both scales. The 16 sectors relate to each other in the same way that the 16 cells within a single sector relate to each other.
This means you can visualise any 4D win line as the combination of two familiar 2D patterns — one at the macro level (sectors) and one at the micro level (cells within sectors).
For example: if the macro pattern is a horizontal line (sectors in the same w-row, z = 1,2,3,4) and the micro pattern is a diagonal (x=1,y=1 → x=4,y=4), then the four cells form a valid 4D win line.
Hover a sector to see the (z,w) position
The 520 Win Axes
There are exactly 520 distinct directions along which a 4-in-a-row line can form:
| Type | Axes that change | Count | Nexus pts |
|---|---|---|---|
| Orthogonal | 1 of 4 axes varies | 256 | 1 |
| Face diagonal | 2 axes vary together | 192 | 2 |
| Space diagonal | 3 axes vary together | 64 | 4 |
| Hyper-diagonal | All 4 axes vary together | 8 | 10 |
| Total | 520 | ||
The 8 hyper-diagonals are the most powerful lines — they pass through all 4 dimensions simultaneously and are very hard to block.
See a live example of each axis type:
Hyper-Nexus Scoring
In Nexus mode, each player places exactly 24 stones. When all 48 stones are placed, every completed line on the board is counted and scored:
- A single stone can be part of many lines simultaneously.
- The player with the highest total score wins.
- There is no elimination — the board always fills completely.
Tip: crossing lines share stones. Building a cluster of stones at the centre gives access to the most axes — but your opponent can score on those lines too.
Hyper-Gravitation
Hyper-Gravitation adapts Connect-4 to four dimensions. Gravity pulls all stones downward along the y-axis within each sector.
On your turn you choose a column — defined by three coordinates (x, z, w). There are 64 such columns. The stone drops to the lowest unoccupied row (y = 4) in that column.
Win condition: same as Hyper-Tic-Tac-Toe — 4 in a row along any of the 520 axes. Because gravity fills from the bottom, diagonal 4D lines become the primary strategic target.
Tactic: watch the column heights in neighbouring sectors (same x, adjacent z or w). Gravity creates natural staircase patterns that lead to diagonal wins across sectors.