SEAM

Quotient Symmetry in Information Geometry

Stochastic Eversion of Antipodal Manifolds

SEAM is an interactive pedagogical instrument for understanding quotient symmetry and information geometry through the visualization of the real projective plane (ℝP²).

The app demonstrates one foundational idea:

When orientation information is discarded, distinct states become identical in a structured, unavoidable way.

SEAM shows this process explicitly by allowing users to apply projective identification—the rule
u ≡ −u—to ordinary geometric objects and observe which distinctions collapse.

This identification is the mathematical foundation from which non-orientability, seams, and ambiguity later arise.

Live Demo

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What SEAM Is (Current Version)

SEAM is not a shape-morphing tool and not a gallery of exotic surfaces.

It is an information-theoretic instrument that cleanly separates:

• State space — what distinctions exist before identification
• Identification rules — which distinctions are forgotten

and lets users explore the consequences of applying those rules.

The current version focuses exclusively on projective identification and the emergence of ℝP².

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Core Mathematical Concept

Direction Space and Quotienting

• Directions in 3D space live on the unit sphere S²

• If orientation is ignored, opposite directions are identified:

  u ≡ −u

• The quotient space formed by this identification is the real projective plane:

  ℝP² = S² / (u ≡ −u)

SEAM visualizes this quotient operationally, not symbolically.

The right panel of the app represents this identification rule directly.

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Interface Overview

SEAM is organized into two panels with strictly distinct roles.

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Left Panel — State Space

Label:
STATE SPACE (ℝ² or ℝ³)

The left panel displays a single base object representing distinct states before identification.

Supported objects include:

Planar Domains (ℝ² embedded in ℝ³)

• Circle
• Triangle
• Square

Spatial Objects (ℝ³)

• Sphere
• Cube
• Pyramid

These objects are:

• Fully orientable
• Continuous (no gaps, cuts, or seams)
• Passive until acted upon by identification

The topology of the object never changes.

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Right Panel — Projective Identification Instrument

Label:
DIRECTION SPACE S²
IDENTIFICATION: u ≡ −u
QUOTIENT: S²/(±) = ℝP²

The right panel is a fixed instrument that represents:

• The unit sphere of directions S²
• With enforced antipodal identification u ≡ −u

It is not a visualization of the left object.

It exists solely to select projective equivalence classes.

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Antipodal Spotlights

Projective identification is visualized using antipodal spotlights:

• A spotlight is defined by a direction u and aperture angle θ
• A point on the left object is highlighted if it lies within θ of u or −u
• Highlights are always:
  • Symmetric
  • Paired
  • Simultaneous

It is impossible to select u without also selecting −u.

This makes equivalence classes visible without cutting or deforming objects.

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Interaction Model

Right → Left

1. User clicks the sphere in the right panel
2. Direction u is selected
3. Antipodal direction −u is automatically paired
4. Left object highlights all points identified with [u]

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Left → Right

1. User clicks a point on the object
2. The direction from the object’s center is computed
3. Its projective class [u] is selected
4. Antipodal cones appear on the right panel

At no point can a single direction exist without its antipode.

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Adjustable Controls (Current)

All user controls affect representation and clarity, never topology or identification rules.

Geometry & Perception

• Mesh resolution / smoothness
• Edge visibility
• Lighting direction
• Lighting intensity
• Camera zoom (constrained)

Highlighting

• Spotlight aperture (cone width)
• Highlight falloff
• Highlight intensity

Constraints

• Planar objects cannot be freely rotated
• No cuts, gaps, or deformations are possible
• Projective identification cannot be disabled

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What SEAM Does Not Do (By Design)

The current version deliberately avoids:

• Showing ℝP² as a surface
• Allowing non-orientable objects as base inputs
• Morphing objects into projective surfaces
• Showing seams or ambiguity regions
• Introducing dynamics, eversion, or time evolution

Non-orientability is treated as a consequence, not a starting point.

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Pedagogical Goal (V1)

The goal of the current version is for users to internalize one fact:

Quotienting collapses distinctions globally, not locally.

Once this is understood, seams and ambiguity become inevitable rather than mysterious.

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Roadmap — Future Development

The following features are not yet implemented and are intentionally staged.

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V2 — Derived Topology Reveal

Goal: Show what emerges from projective identification.

• Ghosted Möbius strip revealed from circle + antipodal ID
• Ghosted ℝP² reference model revealed from sphere + antipodal ID
• Non-interactive explanatory overlays only
• Clear labeling: “Result of projective identification”

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V3 — Seam Visualization

Goal: Make seams explicit as loci of ambiguity.

• Highlight regions where identification density concentrates
• Visualize overlap of equivalence neighborhoods
• Introduce the concept of a seam as identification stress
• No cuts or surgery

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V4 — Dynamic Identification

Goal: Explore identification under motion.

• Direction sweeps
• Continuity vs ambiguity
• Emergent non-orientability in motion

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V5 — Pedagogical Expansion

Goal: Turn SEAM into a full learning instrument.

• Guided lessons
• Concept checkpoints
• Educator annotations
• Exportable visual states

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Design Philosophy (Pinned)

SEAM is not a shape viewer.
It is an instrument that applies a rule.
Non-orientability is not chosen — it is discovered.

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Status

• Current: Functional V1 projective identification instrument
• Focus: Information geometry and quotient symmetry
• Next: Derived topology and seam emergence

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If you understand this README,
you are already prepared to understand why seams must exist.