# Tables of Cell Statistics Across Resolutions

## Cell counts

We list the number of hexagons and pentagons at each H3 resolution. There are always exactly $12$ pentagons at every resolution.

Res | Total number of cells | Number of hexagons | Number of pentagons |
---|---|---|---|

0 | 122 | 110 | 12 |

1 | 842 | 830 | 12 |

2 | 5,882 | 5,870 | 12 |

3 | 41,162 | 41,150 | 12 |

4 | 288,122 | 288,110 | 12 |

5 | 2,016,842 | 2,016,830 | 12 |

6 | 14,117,882 | 14,117,870 | 12 |

7 | 98,825,162 | 98,825,150 | 12 |

8 | 691,776,122 | 691,776,110 | 12 |

9 | 4,842,432,842 | 4,842,432,830 | 12 |

10 | 33,897,029,882 | 33,897,029,870 | 12 |

11 | 237,279,209,162 | 237,279,209,150 | 12 |

12 | 1,660,954,464,122 | 1,660,954,464,110 | 12 |

13 | 11,626,681,248,842 | 11,626,681,248,830 | 12 |

14 | 81,386,768,741,882 | 81,386,768,741,870 | 12 |

15 | 569,707,381,193,162 | 569,707,381,193,150 | 12 |

## Cell areas

Cell areas are computed with a **spherical** model of the earth using the
authalic radius
given by
WGS84/EPSG:4326.

### Average area in km^{2}

The area of an H3 cell varies based on its position relative to the
icosahedron vertices.
We show the **average** hexagon areas for each resolution.
All pentagons within a resolution have the same area.

Res | Average Hexagon Area (km^{2}) | Pentagon Area* (km^{2}) | Ratio (P/H) |
---|---|---|---|

0 | 4,357,449.416078381 | 2,562,182.162955496 | 0.5880 |

1 | 609,788.441794133 | 328,434.586246469 | 0.5386 |

2 | 86,801.780398997 | 44,930.898497879 | 0.5176 |

3 | 12,393.434655088 | 6,315.472267516 | 0.5096 |

4 | 1,770.347654491 | 896.582383141 | 0.5064 |

5 | 252.903858182 | 127.785583023 | 0.5053 |

6 | 36.129062164 | 18.238749548 | 0.5048 |

7 | 5.161293360 | 2.604669397 | 0.5047 |

8 | 0.737327598 | 0.372048038 | 0.5046 |

9 | 0.105332513 | 0.053147195 | 0.5046 |

10 | 0.015047502 | 0.007592318 | 0.5046 |

11 | 0.002149643 | 0.001084609 | 0.5046 |

12 | 0.000307092 | 0.000154944 | 0.5046 |

13 | 0.000043870 | 0.000022135 | 0.5046 |

14 | 0.000006267 | 0.000003162 | 0.5046 |

15 | 0.000000895 | 0.000000452 | 0.5046 |

*: Within a given resolution, all pentagons have the same area.

### Average area in m^{2}

Here are the same areas, but in m^{2}.

Res | Average Hexagon Area (m^{2}) | Pentagon Area* (m^{2}) |
---|---|---|

0 | 4,357,449,416,078.392 | 2,562,182,162,955.496 |

1 | 609,788,441,794.134 | 328,434,586,246.469 |

2 | 86,801,780,398.997 | 44,930,898,497.879 |

3 | 12,393,434,655.088 | 6,315,472,267.516 |

4 | 1,770,347,654.491 | 896,582,383.141 |

5 | 252,903,858.182 | 127,785,583.023 |

6 | 36,129,062.164 | 18,238,749.548 |

7 | 5,161,293.360 | 2,604,669.397 |

8 | 737,327.598 | 372,048.038 |

9 | 105,332.513 | 53,147.195 |

10 | 15,047.502 | 7,592.318 |

11 | 2,149.643 | 1,084.609 |

12 | 307.092 | 154.944 |

13 | 43.870 | 22.135 |

14 | 6.267 | 3.162 |

15 | 0.895 | 0.452 |

*: Within a given resolution, all pentagons have the same area.

### Hexagon min and max areas

The area of an H3 cell varies based on its position relative to the
icosahedron vertices.
We compute the minimum and maximum values for the **hexagon** areas (excluding
the pentagons) at each resolution, and show their ratio.

Res | Min Hexagon Area (km^2) | Max Hexagon Area (km^2) | Ratio (max/min) |
---|---|---|---|

0 | 4,106,166.334463915 | 4,977,807.027442012 | 1.212276 |

1 | 447,684.201817940 | 729,486.875275344 | 1.629468 |

2 | 56,786.622889474 | 104,599.807218925 | 1.841980 |

3 | 7,725.505769639 | 14,950.773301379 | 1.935248 |

4 | 1,084.005635363 | 2,135.986983965 | 1.970457 |

5 | 153.766244448 | 305.144308779 | 1.984469 |

6 | 21.910021013 | 43.592111685 | 1.989597 |

7 | 3.126836030 | 6.227445905 | 1.991613 |

8 | 0.446526174 | 0.889635157 | 1.992347 |

9 | 0.063780227 | 0.127090737 | 1.992635 |

10 | 0.009110981 | 0.018155820 | 1.992740 |

11 | 0.001301542 | 0.002593689 | 1.992782 |

12 | 0.000185933 | 0.000370527 | 1.992797 |

13 | 0.000026562 | 0.000052932 | 1.992802 |

14 | 0.000003795 | 0.000007562 | 1.992805 |

15 | 0.000000542 | 0.000001080 | 1.992805 |

## Edge lengths

Edge lengths are computed with a **spherical** model of the earth using the
authalic radius
given by
WGS84/EPSG:4326.
Average edge lengths were calculated exactly for resolutions 0 through 6 and
extrapolated for finer resolutions.

Res | Average edge length (Km) |
---|---|

0 | 1281.256011 |

1 | 483.0568391 |

2 | 182.5129565 |

3 | 68.97922179 |

4 | 26.07175968 |

5 | 9.854090990 |

6 | 3.724532667 |

7 | 1.406475763 |

8 | 0.531414010 |

9 | 0.200786148 |

10 | 0.075863783 |

11 | 0.028663897 |

12 | 0.010830188 |

13 | 0.004092010 |

14 | 0.001546100 |

15 | 0.000584169 |

## Appendix: Methodology

*Hexagons have 7 hexagon children. Pentagons have 6 children: 5 hexagons and 1 pentagon.*

### Cell counts

By definition, resolution `0`

has $110$
**hexagons** and $12$ **pentagons**, for a total of $122$ **cells**.

In fact, *every* H3 resolution has exactly $12$ **pentagons**, which are always
centered at the icosahedron vertices; the number of **hexagons** increases
with each resolution.

Accounting for both **hexagons** and **pentagons**,
the total number of **cells** at resolution $r$ is

#### Derivation of the cell count formula

We can derive the formula above with the following steps.

First, let $h(n)$ be the number of
children $n \geq 0$ resolution levels below any single **hexagaon**.
Any **hexagon** has $7$ immediate children, so recursion gives us
that

Next, let $p(n)$ be the number of children $n \geq 0$ resolution levels below
any single **pentagon**.
Any **pentagon** has $5$ hexagonal immediate children and $1$ pentagonal
immediate child.
Thus, $p(0) = 1$ and $p(1) = 6$.

For $n \geq 1$, we get the general recurrence relation

For $n \geq 0$, after working through the recurrence, we get that

using the closed form for a geometric series.

Finally, using the closed forms for $h(n)$ and $p(n)$,
and the fact that (by definition)
resolution `0`

has
$12$ **pentagons** and $110$ **hexagons**,
we get the closed form for the total number of **cells**
at resolution $r$ as

#### Jupyter notebook

A notebook to produce the cell count table above can be found here.

### Cell areas

Cell areas are computed with a **spherical** model of the earth using the
authalic radius
given by
WGS84/EPSG:4326.

The `h3-py-notebooks`

repo
has notebooks for producing the
average cell area table
and the
min/max area table.

### Edge lengths

todo