The Regiment System: Organizing Thousands of Polychora

With over 2,000 known uniform polychora, mathematicians needed a systematic way to organize them. The regiment system, developed by researchers like Jonathan Bowers and Richard Klitzing, groups polychora by their shared skeletal structure.

The Core Insight

Multiple distinct polychora can share the exact same vertices and edges—their wireframe skeleton is identical. What differs is how faces and cells are arranged on that skeleton.

Think of it like this: given the same set of vertices and edges, you can connect them into different 2D faces, and group those faces into different 3D cells, creating entirely different 4D shapes.

Military Terminology

The classification uses military ranks as a memorable hierarchy:

Army

An army is the largest grouping: all polychora that share the same vertex set.

The convex hull of these vertices (the "most convex" shape with those vertices) is called the General
All army members have vertices at the same positions

Example: The 600-cell army contains every polychoron whose 120 vertices match the 600-cell's vertex arrangement.

Regiment

A regiment is a subset of an army: all polychora that share the same vertices AND edges.

The most symmetric member is called the Colonel
Regiment members differ only in their faces and cells
The regiment is named after its colonel

Example: The Ico (24-cell) regiment contains 14 polychora that all have the same 24 vertices and 96 edges as the 24-cell, but different face/cell arrangements.

Company

A company is an even finer grouping: polychora that share the same vertices, edges, AND faces.

Company members differ only in how faces are grouped into cells
The leading member is the Captain

Why Regiments Matter

Shared Properties

Regiment members share:

Vertex count and positions
Edge count and connections
Often similar symmetry properties
The same "wireframe" appearance in projections

Different Properties

Regiment members differ in:

Face types and arrangements
Cell types and counts
Convexity (some members may be non-convex)
Orientability (some may be non-orientable)

Research Applications

The regiment system helps researchers:

Systematically search for new polychora
Understand relationships between shapes
Predict what forms might exist
Organize the massive catalog of known shapes

Notable Regiments

The Ico Regiment (24-cell)

Colonel: Icositetrachoron (24-cell)
Symmetry: F₄ (unique to 4D)
Members: 14 uniform polychora

The Ico regiment is fascinating because it includes:

The convex 24-cell itself
Non-convex members like the Octahemihexadecachoron
Non-orientable members with "hemi" facets
The compound of three tesseracts (Gico)

All 14 shapes have identical wireframes but dramatically different cell structures.

The Ex Regiment (600-cell)

Colonel: Hexacosichoron (600-cell)
Symmetry: H₄
Members: Multiple, including star polychora

The 600-cell's 120 vertices and 720 edges support many arrangements, including several of the 10 regular star polychora.

The Gap Regiment (Grand Antiprism)

Colonel: Grand Antiprism
Members: 2 (the smallest non-trivial regiment)

The Grand Antiprism is unique: it's the only convex uniform polychoron that cannot be generated by the standard Wythoff construction. Its regiment is correspondingly unusual.

OBSA: Bowers Style Acronyms

To manage the unwieldy names of thousands of polychora, Jonathan Bowers developed the OBSA (Official Bowers Style Acronym) system.

Naming Logic

Base names derive from the regular forms:

Pen = Pentachoron
Tes = Tesseract
Hex = Hexadecachoron
Ico = Icositetrachoron
Hi = Hecatonicosachoron
Ex = Hexacosichoron

Prefixes indicate modifications:

s- = Small (e.g., Sidpith)
g- = Great or Grand (e.g., Gahi)
r- = Rectified (e.g., Rit)
t- = Truncated (e.g., Thi)
p- = Prismato- (e.g., Prit)

Examples

OBSA	Full Name	Derivation
Rit	Rectified Tesseract	r + Tes → Rit
Srit	Small Rhombated Tesseract	s + r + Tes → Srit
Grahi	Great Rhombated Hecatonicosachoron	g + r + Hi → Grahi

Pronunciation

OBSA names are designed to be pronounceable:

Rit = "rit" (rhymes with "bit")
Sishi = "shee-shee"
Grahi = "grah-hee"
Sidpith = "sid-pith"

The 2021 Discovery: Idtessids

For 14 years (2006-2020), the count of uniform polychora was stable. Then the Polytope Discord community discovered the Idtessid family—333 new uniform polychora hiding in previously unexplored regiment territory.

This demonstrates that the regiment system isn't just organizational—it's a roadmap for discovery. By systematically exploring regiment skeletons, researchers continue to find new valid polychora.

Special Cases

Fissaries

Some regiment members are fissary: they look like compounds (multiple overlapping shapes) but share elements in ways that make them arguably single entities.

Exotics

Exotic polychora have coinciding cells—multiple cells occupying the same 3D space. These are mathematically valid but usually excluded from "standard" uniform counts.

Non-Wythoffian Forms

Most uniform polychora can be generated by the Wythoff construction (reflection groups and kaleidoscopes). The Grand Antiprism is the only convex uniform that cannot—making its regiment uniquely interesting.

Using This Knowledge

When exploring polychora in the viewer:

Regiment members share the same wireframe—watch for this when toggling between related shapes
OBSA names tell you about a shape's derivation—"Rit" is rectified, "Srit" adds rhombation
Category numbers group similar construction methods
Army membership tells you which shapes share vertices

Understanding regiments transforms the vast catalog from an overwhelming list into a structured map of 4D geometry.