disk covering problem

{{unsolved|mathematics|What is the smallest real number r(n) such that n disks of radius r(n) can be arranged in such a way as to cover the unit disk?}}

The disk covering problem asks for the smallest real number r(n) such that n disks of radius r(n) can be arranged in such a way as to cover the unit disk. Dually, for a given radius ε, one wishes to find the smallest integer n such that n disks of radius ε can cover the unit disk.{{citation

| last = Kershner | first = Richard

| journal = American Journal of Mathematics

| mr = 0000043

| pages = 665–671

| title = The number of circles covering a set

| volume = 61

| year = 1939

| issue = 3

| doi=10.2307/2371320| jstor = 2371320

}}.

The best solutions known to date are as follows.{{cite web |url=https://erich-friedman.github.io/packing/circovcir/|title=Circles Covering Circles|last=Friedman|first=Erich|access-date=4 October 2021}}

class="wikitable" border="1"
n

! r(n)

! Symmetry

1

| 1

| All

2

| 1

| All (2 stacked disks)

3

| \sqrt{3}/2 = 0.866025...

| 120°, 3 reflections

4

| \sqrt{2}/2 = 0.707107...

| 90°, 4 reflections

5

| 0.609382... {{OEIS2C|A133077}}

| 1 reflection

6

| 0.555905... {{OEIS2C|A299695}}

| 1 reflection

7

| 1/2 = 0.5

| 60°, 6 reflections

8

| 0.445041...

| ~51.4°, 7 reflections

9

| 0.414213...

| 45°, 8 reflections

10

| 0.394930...

| 36°, 9 reflections

11

| 0.380083...

| 1 reflection

12

| 0.361141...

| 120°, 3 reflections

Method

The following picture shows an example of a dashed disk of radius 1 covered by six solid-line disks of radius ~0.6. One of the covering disks is placed central and the remaining five in a symmetrical way around it.

File:DiscCoveringExample.svg

While this is not the best layout for r(6), similar arrangements of six, seven, eight, and nine disks around a central disk all having same radius result in the best layout strategies for r(7), r(8), r(9), and r(10), respectively. The corresponding angles θ are written in the "Symmetry" column in the above table.

References

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