A383461 Number of vertices in graph G_n formed by taking a regular n-gon with all its chords extended to infinity (the n-th graph in A344857) and inverting it in its circumscribing circle.

4, 5, 16, 37, 92, 145, 334, 471, 892, 901, 1964, 2185, 3796, 3969, 6682, 5563, 10964, 11141, 17032, 17293, 25324, 21913, 36326, 36479, 50572, 50485, 68644, 51661, 91172, 90753, 118834, 118355, 152356, 139861, 192512, 191445, 240124, 238481

Offset: 3

Author

Scott R. Shannon and N. J. A. Sloane, Jun 02 2025

Keywords

• nonn

Comments

Inverting a point or a line in a circle C with center O and radius r is a classical operation in geometry (Coxeter, Section 6.3; Pedoe, pp. 4-9). Every point A inside C except O itself has an inverse point A' outside the circle; A' lies on the line OA and satisfies |OA|*|OA'| = r^2. The inverse of the center O is undefined.

If a line L passes through O its inverse is L itself. If L is not a diameter of C, and meets C in two points A and B, the inverse of L is the circle through O, A, and B.

Theorem: G_n has A345025(n) regions. If n is even then n of these regions are infinite, otherwise there is a single infinite region.

The initial versions of the illustrations were made by NJAS using GeoGebra. The colored versions were added later by SRS using a Java program. These have greater resolution and include information about the vertex and region counts.

References

• H. S. M. Coxeter, Introduction to Geometry, Wiley, 1961.
• D. Pedoe, Circles: A Mathematical View, Dover, 1979.

Links

• Scott R. Shannon, The graph G_13 consists of 78 circles. There are a(13) = 1964 vertices.
• Scott R. Shannon, The graph G_13 (continued). There are 2172 regions (2171 finite regions and one infinite region).
• N. J. A. Sloane, The graph G_3 consists of three circles. There are a(3) = 4 vertices and 7 regions (6 finite regions and one infinite region).
• N. J. A. Sloane, GeoGebra source file for G_3
• N. J. A. Sloane, The graph G_4 consists of two lines and four circles. There are a(4) = 5 vertices and 16 regions (12 finite regions and 4 infinite regions).
• N. J. A. Sloane, GeoGebra source file for G_4
• N. J. A. Sloane, The graph G_5 consists of ten circles. There are a(5) = 16 vertices and 36 regions (35 finite regions and one infinite region).
• N. J. A. Sloane, GeoGebra source file for G_5
• N. J. A. Sloane, The graph G_6 consists of three lines and 12 circles. There are a(6) = 37 vertices and 72 regions (66 finite regions and 6 infinite regions).
• N. J. A. Sloane, GeoGebra source file for G_6
• N. J. A. Sloane, The graph G_7 consists of 21 circles (colored red). There are a(7) = 92 vertices and 141 regions (140 finite regions and one infinite region). See following illustration for an enlargement of the central heptagonal portion of the graph. Note that the blue heptagon is not part of G_7.
• N. J. A. Sloane, An enlargement of the central heptagonal portion of the previous illustration, showing the 91 individual cells more clearly.
• N. J. A. Sloane, GeoGebra source file for G_7
• N. J. A. Sloane, The graph G_8 consists of four lines and 24 circles. There are 145 vertices and 232 regions (224 finite regions and 8 infinite regions).
• N. J. A. Sloane, GeoGebra source file for G_8

Crossrefs

Cf. A146212, A344857, A345025, A383462.

Formula

a(n) = A146212(n) + (n mod 2).