A flat torus is represented by a rectangle whose opposite edges are identified: left with right, and top with bottom. Crossing one colored seam returns you at the matching position on its partner without changing orientation. The rectangle is a coordinate chart, not a physical wall.
Top ↔ BottomLeft ↔ Right
YouGhost entryLeft ↔ RightTop ↔ Bottom
Play the seam, not the border
In Torus mode, both pairs of opposite edges wrap. If the blue explorer token leaves through the red right edge, it appears at the same height on the red left edge. Crossing cyan at the top returns it at the matching horizontal position along the cyan bottom. Its direction stays unchanged, which is why motion looks continuous even though the chart jumps.
Collect twelve seeds before the 75-second timer ends. Bramble hazards move on the same surface and cross the same seams. A collision costs one of three lives; a short recovery window prevents one overlap from removing several lives at once.
What changes on a Möbius strip
A Möbius strip can be represented by a rectangle with one pair of opposite edges identified in reverse. Seamshift joins the red left and right edges this way. Cross the red seam at height y and return at height 1 − y; vertical direction also reverses. The cyan top and bottom are true boundaries, so the token reflects there instead of wrapping.
This is a deliberately narrow comparison, not a claim that every “twisted world” game uses identical coordinates. Switching modes before a run makes the changed rule visible using the same controls, field, seeds, and hazards.
Ghosts make edge identification visible
When the explorer nears a joined edge, a dashed ghost appears at its equivalent entry position. On the torus, a ghost beside the opposite red or cyan seam stays aligned. On the Möbius strip, the red-seam ghost appears at the reflected height. These previews show the topological rule before the actual crossing.
The edge colors are reinforced by labels and ghost styles, so color is not the only cue. The current mode and a brief rule sentence remain above the field. Seam-cross events are also announced in the game's status message.
How to play Seamshift Arena
Choose Torus or Möbius Strip while the game is idle. Press Start 75s Run, then move with WASD, arrow keys, or the four large touch buttons. Gather golden winged seeds and red berries. Avoid thorny burrs and dry bramble knots. The result screen reports mode, collected seeds, lives, seam crossings, and score.
Replay keeps the selected surface so the same rule can be practiced again. Return to the mode controls to compare the other surface. The separate How to Play page gives exact keyboard, touch, scoring, and wrap rules.
A mathematical game, not a shooter
The clean-room design uses a botanical field notebook, a round explorer token, seed collectibles, and plant hazards. It does not use a spacecraft, rocks, weapons, a starfield, or the title and trade dress of any arcade shooter. The mapping functions and rules were implemented independently from the mathematical definitions stated on this page.
Torus game questions
Is a torus the same as a wraparound rectangle?
The rectangle is a convenient coordinate representation. Identifying both pairs of opposite edges produces the topology of a torus even though the screen remains visually flat.
Why does the Möbius token flip vertically?
The joined red edges use reversed parameter order. A point at height y returns at 1 − y, and the vertical component of its direction reverses.
Is Seamshift Arena free?
Yes. Both surfaces, the complete 75-second loop, touch controls, results, and replay are available without payment or registration.