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Hexagon Pie?

I discovered these "isometric" grids (Dot's right!) when some of my students took on a project which involved sketching structures made from cubes. Do you see a cube in the grid above, or a hexagon with three spokes? To live in the visual world created by these grids is to live in two- and three dimensions at once. [Covid update: an isogrid arrangement of seats in an open space will distance more people safely.]

Playtime First!

Before I attempted anything technical, I had a little sketching fun. Visit my Iso-Gallery – the product of a pleasant hour on a warm spring afternoon.

The Geometry of the Grid

Let's start with angles – they're easy. Each small triangle is regular, so it has three angles of 60 degrees each. The rhombuses, made by joining two triangles, have pairs of 60- and 120-degree angles.

The lengths of the figures need not be complicated either – that is, if you are willing to accept a new perspective in this non-rectangular landscape. Let the smallest distance between any two dots be our unit. That makes the sides of the smallest triangles and rhombuses measure one each. What's new is that we will call the heights of these same figures one as well. If you insist on calculating the height vertically (as you were taught to do in school), then the height works out to one-half the square root of three. Don't go there!

We reap benefits from this new "slant" on height. Perimeters are not affected – the triangle's is three and the rhombus's four, but say(!), the area formulas work too. The triangle's area is base times height divided by two, or an area of one-half of a rhombus. The rhombus has an area of base times height, or one unit in the new system. We won't call it a square unit, though. Since there are no squares around to become jealous, we allow the rhombus to be our unit of area. Hexagon π

Fractions would be so much simpler if pies were baked in hexagonal rather than circular tins. We would have to understand only halves, thirds, and sixths. I've never had the desire (or will) to cut a homemade pecan pie into more than six slices, have you? Perhaps we could allow fourths and eighths on special days, and we'll have more on them later.

When using the isogrids to relate hexagons and circles, I discovered a new mathematical constant. I call it Hexagon π. Readers should be familiar with the traditional π as the constant ratio of the circumference of any circle to its diameter. Humor me for a second and look at the hexagon as a circle. It's the best we can do if we stick to unit segments connecting grid points.

If hexagons on the grid are to preserve the circle formulas for circumference (2 x π x r) and area (π x r x r), then "hexagon π" must equal 3. In the figure, you see three nested hexagons with radii measuring 2, 3, and 4. In the smallest hexagon, we see a perimeter of 12, and if 2 x π x r is 12, then π must be 3. In that same hexagon, we count rhombuses to arrive at an area of 12 (pi x r x r), and again π works out to be 3. The other measures are presented in the iso-table at the bottom. Fourths and Eighths

The reader can easily verify, using the hexagon of radius 2 above, that a hexagon of radius one, centered inside, has an area of 3 (rhombuses) and is therefore a one-fourth part. To share a hexagon pie (radius 2) among four people, this division requires someone who doesn't like crust. Can the same hexagon (radius 2) be divided into equal eighths that all have the same shape? See Solution below after you've thought about it.

April 2014, October 2021 