# Why hexagons?

The aggregation of incident point data to regularly shaped grids is used for many reasons such as normalizing geography for mapping or to mitigate the issues of using irregularly shaped polygons created arbitrarily (such as county boundaries or block groups that have been created from a political process). Regularly shaped grids can only be comprised of equilateral triangles, squares, or hexagons, as these three polygon shapes are the only three that can tessellate (repeating the same shape over and over again, edge to edge, to cover an area without gaps or overlaps) to create an evenly spaced grid.

Though the square (fishnet) grid is the predominantly used shape type in GIS analysis and thematic mapping, there are ways in which hexagons may be better suited for your analysis based on the nature of your question.

Reasons to consider aggregating into a hexagon grid are the following:

• Hexagons reduce sampling bias due to edge effects of the grid shape, this is related to the low perimeter-to-area ratio of the shape of the hexagon. A circle has the lowest ratio but cannot tessellate to form a continuous grid. Hexagons are the most circular-shaped polygon that can tessellate to form an evenly spaced grid.
• This circularity of a hexagon grid allows it to represent curves in the patterns of your data more naturally than square grids.
• When comparing polygons with equal areas, the more similar to a circle the polygon is, the closer to the centroid the points near the border are (especially points near the vertices). This means that any point inside a hexagon is closer to the centroid of the hexagon than any given point in an equal-area square or triangle would be (this is due to the more acute angles of the square and triangle versus the hexagon).
• Hexagons are preferable when your analysis includes aspects of connectivity or movement paths.
• Due to the linear nature of rectangles, fishnet grids can draw our eyes to the straight, unbroken, parallel lines which may inhibit the underlying patterns in the data. Hexagons tend to break up the lines and allow any curvature of the patterns in the data to be seen more clearly and easily. This breakup of artificial linear patterns also diminishes any orientation bias that can be perceived in fishnet grids.
• If you are working over a large area, a hexagon grid will suffer less distortion due to the curvature of the earth than the shape of a fishnet grid.
• Finding neighbors is more straightforward with a hexagon grid. Since the edge or length of contact is the same on each side, the centroid of each neighbor is equidistant. However, with a fishnet grid, the Queen's Case (above/below/right/left) neighbor's centroids are N units away, while the centroids of the diagonal (Rook) neighbors are farther away (exactly the square root of 2 times N units away).
• Since the distance between centroids is the same in all six directions with hexagons, if you are using a distance band to find neighbors or are using the Optimized Hot Spot Analysis, Optimized Outlier Analysis or Create Space Time Cube By Aggregating Points tools, you will have more neighbors included in the calculations for each feature if you are using hexagonal grid as opposed to a fishnet grid.