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16-cell: Geometry, Overview & Symmetry constructions

In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid .

Language: English [EN]
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16-cell topic overview

The analysis highlights Geometry, Overview and Symmetry constructions as prominent areas in the source structure around 16-cell.

Related topics
89
Source areas
8
Connected nodes
97
Extracted relationships
75
Concept neighborhoods
39
Bridge connections
97

What this topic covers Research coverage

Source areas are shown by the number of related topics found in each part of the analysis. Use smaller areas too: they can reveal specialized angles and content gaps.

Overview · 39 topics
Geometry · 21 topics
Related uniform polytopes and honeycombs · 7 topics
Symmetry constructions · 7 topics
Tessellations · 6 topics
Projections · 5 topics
4 sphere Venn diagram · 2 topics
Related complex polygons · 2 topics

Smaller areas are not necessarily less important. They contain fewer connections in this analysis and can be useful for finding specialized angles or coverage gaps.

Key facts & relationships

High-confidence facts extracted from structured source data. Use them as anchors for further research.

Cells
16 {3,3}
Coxeter group
B4, [3,3,4], order 384 D4, order 192
Dual
Tesseract
Edges
24
Faces
32 {3}
Petrie polygon
octagon

Explore all related topics Closing gaps

Browse the complete topic structure, not only the most central items. Less prominent entities and concepts can reveal missing angles, specialized context and useful research gaps. Each item opens a new analysis centered on that subject.

Overview

Geometry

Tessellations

Projections

4 sphere Venn diagram

Symmetry constructions

Related complex polygons

Related uniform polytopes and honeycombs

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

How 16-cell connects Entity context

The extracted context around 16-cell shows recurring relationship patterns in the source. For example, 16-cell → Archived, Cell, Der, Description, Eric, German, Marco Möller's Regular, MathWorld, R4, Richard, Wayback MachineKlitzing, Weisstein, Zeller Another extracted example is 16-cell → Each, Euclidean, Hence, One, R4, Schläfli, The, This, Together, Twenty-four. Use these groups to spot repeated connection types before inspecting the individual relationships.

16-cell

Top relations

related to External links · 13
16-cell → Archived, Cell, Der, Description, Eric, German, Marco Möller's Regular, MathWorld, R4, Richard, Wayback MachineKlitzing, Weisstein, Zeller
related to Tessellations · 10
16-cell → Each, Euclidean, Hence, One, R4, Schläfli, The, This, Together, Twenty-four
related to Structure · 6
16-cell → At, It, Its, The, The Schläfli, There
is a · 5
16-cell → 4-dimensional cross polytope, regular convex 4-polytope, second in the sequence of 6 convex regular 4-polytopes, simple frame in which to observe 4-dimensional rotations, simplest regular polytope in which they occur
related to Related complex polygons · 5
16-cell → It, Its, Kantor, The, The Möbius
related to Rotations · 5
16-cell → Completely, Euclidean, In, Rotations, The
related to Symmetry constructions · 5
16-cell → B4, Coxeter, It, The, There
related to Coordinates · 3
16-cell → All, Cartesian, The
related to Projections · 3
16-cell → In, Surrounding, The
related to Related uniform polytopes and honeycombs · 3
16-cell → B4, D4, The

Important terminology

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

vertices regular edges orthogonal vertex three cells tetrahedral one two squares symmetry octahedron planes tetrahedra completely tetrahedron also great rotation

16-cell relationships Subject–Predicate–Object triples

TTTA extracted 75 structured relationships around 16-cell. Examples in this analysis include 16-cell → Cells → 16 {3,3} and 16-cell → Coxeter group → B4, [3,3,4], order 384 D4, order 192. The table shows each extracted connection, where it came from and its confidence.

SubjectPredicateObjectConfidenceSrc
16-cellCells16 {3,3}1.00infobox
16-cellCoxeter groupB4, [3,3,4], order 384 D4, order 1921.00infobox
16-cellDualTesseract1.00infobox
16-cellEdges241.00infobox
16-cellFaces32 {3}1.00infobox
16-cellPetrie polygonoctagon1.00infobox
16-cellPropertiesconvex, isogonal, isotoxal, isohedral, regular, Hanner polytope1.00infobox
16-cellSchläfli symbol{3,3,4}1.00infobox
16-cellTypeConvex regular 4-polytope 4-orthoplex 4-demicube1.00infobox
16-cellUniform index121.00infobox
16-cellVertex figureOctahedron1.00infobox
16-cellVertices81.00infobox
16-cellis aregular convex 4-polytope0.90text
16-cellis asecond in the sequence of 6 convex regular 4-polytopes0.90text
16-cellis asimple frame in which to observe 4-dimensional rotations0.90text
16-cellis asimplest regular polytope in which they occur0.90text
16-cellis a4-dimensional cross polytope0.90text

Related concept clusters Concept neighborhoods

The concept neighborhoods around 16-cell bring nearby vocabulary together. In this analysis, examples include Regular, Vertices and Three. Use the clusters to find adjacent concepts and terminology that may deserve separate research.

  • 16-cell
    • Regular
    • Vertices
    • Three
    • Vertex
    • Orthogonal
    • Symmetry
    • Edges
    • Cells
    • Tetrahedral
    • One
    • Schläfli
    • Tesseract
  • 16-cell
    • Regular
    • Vertices
    • Three
    • Vertex
    • Orthogonal
    • Symmetry
    • Edges
    • Cells
    • Tetrahedral
    • One
    • Schläfli
    • Tesseract
  • regular convex 4-polytope
    • 4-polytopes
    • Schläfli
    • Characteristic
    • Tesseract
    • 5-cell
    • Polytope
    • Cells
    • Polytopes
    • Regular
    • Tetrahedral
    • Tetrahedron
    • 4-dimensional
  • characteristic tetrahedron of the regular tetrahedron
    • 5-cell
    • Tetrahedron
    • 4-polytopes
    • Characteristic
    • Regular
    • Cells
    • Tetrahedral
    • Polytopes
    • Schläfli
    • 4-dimensional
    • Tetrahedra
    • Three
  • regular tetrahedra
    • 4-polytopes
    • Characteristic
    • 5-cell
    • Cells
    • Tetrahedral
    • Polytopes
    • Tetrahedron
    • Schläfli
    • Two
    • 4-dimensional
    • Different
    • Ring
  • regular octahedron
    • 4-polytopes
    • Characteristic
    • 5-cell
    • Cells
    • Tetrahedral
    • Polytopes
    • Tetrahedron
    • Octahedral
    • Schläfli
    • Polytope
    • 4-dimensional
    • Tetrahedra
  • cells
    • Tetrahedral
    • Tetrahedra
    • Regular
    • Tetrahedron
    • Two
    • Dual
    • Schläfli
    • Tesseract
    • Vertices
    • Octahedron
    • Edges
    • Vertex
  • edges
    • Vertices
    • Vertex
    • Squares
    • Eight
    • Pairs
    • Tetrahedron
    • Opposite
    • One
    • Regular
    • Two
    • Orthogonal
    • Octahedral

Connections between topic areas Semantic bridges

For 16-cell, one of the stronger structural bridges in this analysis connects 16-cell with Overview. Bridges highlight paths between different parts of the map and can reveal research angles that are easy to miss in a flat list.

Min side: 3
16-cell — Overview · splits 58 ⟂ 40
16-cell — Geometry · splits 76 ⟂ 22
16-cell — Symmetry constructions · splits 90 ⟂ 8
16-cell — Related uniform polytopes and honeycombs · splits 90 ⟂ 8
16-cell — Tessellations · splits 91 ⟂ 7
16-cell — Projections · splits 92 ⟂ 6
16-cell — 4 sphere Venn diagram · splits 95 ⟂ 3
16-cell — Related complex polygons · splits 95 ⟂ 3

Map overview Semantic statistics

16-cell

Nodes98
Edges97
Triples75
Avg. degree1.98
Density0.020408
Components1

Source & methodology

TTTA analyzes the structure around 16-cell to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as Geometry, Overview & Symmetry constructions, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.

Source: Wikipedia — 16-cell · EN edition · Analysis: TopicsToTalkAbout

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