The renderer rests, the topology holds: one surface, no inside, and the ring where the neck passes through the wall is still only a scar of projection.
- FIG-8 IMMERSION K → ℝ³
- GLASS · ONE SIDE
- MÖBIUS LAP
- 01
- ORIENTATION
- ⊕ ORIGINAL
- u
- 2.35 RAD
◌ SELF-INTERSECTION
not part of the bottle — a scar of projection. in ℝ⁴ the neck passes beside the wall and the glass never touches itself.
FORM · TOPOLOGY LAB · ONE SURFACE, NO EDGE
The bottle
with no inside.
A Klein bottle holds nothing, on principle. Pour it full and it pours itself out — the glass has one side, and wherever you stand on it, you are already outside.
01 · SEAM
No edge, no rim, no verdict
Run a fingertip anywhere on the glass. You will never cross an edge, never climb a rim, never find the moment you moved from outside to in — there is no such moment. A sphere settles the question at every point: this side water, that side world. The Klein bottle refuses to rule. Its inside is its outside, continued.
Build one by hand: take a cylinder, and instead of gluing its ends into a torus, pass one end through the wall and glue it from within, reversed. The seam closes perfectly. The distinction between the two faces of the glass does not survive the trip.
χ = 0 · non-orientable · genus 2 (non-orientable) · immersed, never embedded, in ℝ³
Pour it full and it pours itself out.
EVERY INTERIOR IS A RUMOUR
02 · TRANSIT
Proof by transit
Beside the bottle, a rehearsal. An arrow walks the midline of a Möbius strip, its flag standing straight off the surface. One full lap — no jump, no flip, no sleight of wrist — and it comes home mirrored, flag pointing through the floor it left from. Walk it once more and it rights itself. The readout above the glass keeps the score.
That is what one-sided means, said without symbols: the far side of the paper is the near side, further along. Cut the Klein bottle down its plane of symmetry and it falls into two of these strips — the bottle is the Möbius strip's alibi, sealed.
orientation-reversing loop · the flag's frame returns as its own reflection · w₁ ≠ 0
03 · FOOTNOTE
The honest footnote
The faint ring where the surface passes through itself is not part of the bottle. It is a scar of projection — the fee ℝ³ charges for displaying a surface that only closes cleanly in four dimensions. Grant one more axis and the neck slides past the wall the way a bridge clears a river that looks, on the map, like a crossing.
The true Klein bottle never touches itself. What glows here is a shadow of the shortage of room.