SpatialHexGeometry Learning

Geometry Learning

The same three tiles can have different outlines

Use three equal regular hexagons labelled A, B and C, each with side length one. Tiles do not overlap and contacts are along complete edges. Begin by counting six edges for every tile. Each shared edge appears twice in that starting count, once for each tile touching it, and neither copy is exposed. Subtract two for each shared edge. The following two fictional layouts have no holes, so their exposed-edge count equals their outer boundary length in unit sides.

Three hexagons in a straight chain in the SpatialHex exposed-edge worksheetStraight chain · contacts AB and BCABC18 starting edges − 2 × 2 shared edges14 exposed unit edges

01

Trace a straight chain with two shared edges

Arrange A, B and C in a straight chain. A shares one edge with B, and B shares one different edge with C; A and C do not touch. Three tiles contribute eighteen edges before contacts are removed. The two shared edges remove four counted edges, leaving fourteen exposed edges. The contact list is simply AB and BC. Tracing the outside provides an independent way to confirm the fourteen-unit outline. Do not subtract an edge merely because two labels appear near one another: the diagram must show the complete edge contact that the list describes.

Three hexagons in a compact cluster in the SpatialHex exposed-edge worksheetCompact cluster · AB, BC and CAABC18 starting edges − 2 × 3 shared edges12 exposed unit edges

02

Compare a compact cluster with three shared edges

Now arrange the same three tiles around a common vertex so that each pair shares an edge. The contact list is AB, BC and CA. There are still eighteen edges in the starting count, but three shared edges remove six, leaving twelve exposed edges. This compact cluster has two fewer exposed edges than the chain even though both contain three tiles. The result concerns boundary length under the stated geometry. It does not say that one arrangement is better for a particular game, pattern or physical object; those choices depend on additional aims.

The exposed-edge counting rule in the SpatialHex exposed-edge worksheetExposed edges = 6 × tiles − 2 × contactsCount complete shared edges onceCorner touches remove no edge lengthBoth example layouts have no holesWith holes, include their exposed boundaries

03

Save the contact list with the boundary calculation

A reusable sheet records tile count, side length, contact pairs, shared-edge count and exposed-edge total. For these unit tiles the rule is six times the tile count minus twice the number of shared edges. Store the sketch beside the contact list so an impossible or missing contact can be spotted. If a larger arrangement contains holes, the same subtraction counts all exposed boundaries, including hole boundaries; it no longer gives only the outer outline. Record that distinction before applying the worksheet to a more elaborate pattern, and keep corner-only contacts outside the shared-edge count.

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