Contact Dimension Calculator

touching spheres dimension

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About Contact Dimension Calculator

A contact dimension calculator computing cd(G): minimum d such that G = contact graph of non-overlapping d-spheres (touching = adjacent). Penny graphs: d=2. Apollonian networks. cd differs from intersection (touching not overlapping). Client-side.

Contact Dimension Calculator Features

  • cd(G)
  • Touching
  • Penny d=2
  • Packing
  • Common graphs
Contact dimension cd(G): minimum d for G = contact graph of non-overlapping d-spheres. Vertices are spheres; edges when spheres touch (tangent). Penny graphs (d=2, equal radii). Koebe-Andreev-Thurston: planar iff contact graph of 2D disks.

How to Use

Select graph:

  • cd: Contact dim
  • Touching: Not overlap
  • Penny: d=2 equal

Circle Packing Theorem

Koebe-Andreev-Thurston: every planar graph is a contact graph of circles in 2D! So cd(G) ≤ 2 for planar. This beautiful theorem connects planar topology to sphere packing geometry.

Higher Dimensions

d=3: sphere packing contact graphs. Not all graphs are contact graphs in any fixed d. Apollonian gaskets produce interesting contact graphs. Related to sphere packing density problems.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Compute cd.
  3. 3Check planarity.
  4. 4Apply Koebe.
  5. 5Visualize packing.

Contact Dimension Calculator — Frequently Asked Questions

Contact vs intersection graphs?+

Intersection: spheres overlap → adjacent. Contact: spheres just touch → adjacent. Contact is much more restrictive. Intersection allows arbitrary overlap; contact requires tangency (no overlap).

What's the circle packing theorem?+

Every planar graph is the contact graph of circles! Proven by Koebe (1936), extended by Andreev and Thurston. The representation is unique for triangulations (up to Möbius). Profound connection.

What are penny graphs?+

Contact graphs of equal-radius circles in 2D. Named after pennies on a table. Always planar, always ≤ 6-regular (hexagonal packing). Recognition is NP-hard despite simple geometric definition.

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