Rectagon

This Idea really needs a Computer:Game:Tools category. That's because when laying out the graphics for a game, this particular rectangle can be a useful tool.

Let's start with the specific and "regular" rectangle, also known as a "square". In games like dungeon adventures, squares are the common building block for the floor plans of the dungeons. That's mostly because it is so conventional to build walls at 90-degree angles that almost no one does it any other way.

Now while a grid of squares can cover the entirety of a flat surface ("tile the plane"), in terms of movement from one square to another, it is well known that moving diagonally lets you cover more distance than moving orthogonally, when whole squares are used as distance-measuring elements. Inside a structure, however, this is not usually a problem, because walls tend to prevent large diagonal movements.

Out in open fields, though, the walls aren't there, and so mapping tends to be done using hexagon grids. Hexagons can also tile a plane, and they offer more accurate distance-measurements, when moving at an angle. The center-to-center distance between a hexagon and each of its neighbors is the same (not true for a grid of squares).

The problem I wish to point out is that when developing games on a computer, the viewscreen is, at its finest scale, a grid of squares. It is difficult to create a grid of really smooth-edged hexagons on such a background-grid of squares --see link for an example that you could load into an image editor, and expand until you can see the pixels making up the hex grid. For that particular grid, I did not need "regular" hexagons (every side the same length, and every angle the same angle), but any serious game-developer could very easily need accurately-drawn regular hexagons.

The Rectagon is a proposed compromise solution to the problem. It has a very particular ratio of height to width --any other ratio is "just" an ordinary rectangle (except when it is a square).

To see how the Rectagon can be derived from a hexagon, consider the following ASCII sketch:
/|\ (pretend the lines meet at a point)
|-| (ignore the hyphen for the moment)
\|/ (pretend the lines meet at a point)
I added the interior vertical bars so that you can imagine some triangles (horizontal lines not provided in the sketch). If the upper two triangles are clipped off, they could be meshed against the lower two triangles (upper-left against lower-right, and upper-right against lower-left), and the result IS the Rectagon.

If that was done for every hexagon in a regular grid, the result would be a grid of Rectagons, and the key feature of the hex grid would be retained (same center-to-center distance between neighbors). It would look a great deal like an ordinary brick wall, except that bricks don't normally have the critical height- to-width ratio.

We can now compute that critical ratio, since we know exactly how to derive a Rectagon from a regular hexagon, which we assume every side had a length of ONE unit (any unit).

From the center of the hexagon to its upper corner is a distance of 1 unit (ditto to the bottom corner). If we now reference the hyphen in the sketch, and imagine it as a line going all the way across the middle of the hexagon, then we can see that since the lengths of the vertical sides are also 1 unit, the distance from the center to the place where we clip off the upper triangles is 1/2 unit. SO, when we move the two triangles to the bottom of the hexagon, we have that full unit of distance from the center to the bottom, plus that 1/2 unit of distance from center to the upper cut-off place, which means that the height of the Rectagon is exactly 1.5 units.

From the center to the upper-right corner is also 1 unit (note that sketch is not a REGULAR hexagon), and since the "drop" from that corner to the middle of the side, where the hyphen-line bisects it, is 1/2 unit, we have two sides of a right-angle triangle that lets us compute the third side (via Pythagorean Theorem) as being 1/2 of the square root of 3 (or "(sqrt(3))/2"), the distance from the center to the side. That means the full width of the hexagon (twice the computed distance) is sqrt(3) units, about 1.732 units. The Rectagon certainly has that width.

Thus the height-to-width ratio of a Rectagon is 1.5 to sqrt(3), exactly that ratio, and no other ratio. It is a unique rectangle. When we imagine every hexagon in a regular grid getting converted to a Rectagon in the manner described above, we note that the center-point of EVERY single hexagon needs to move down exactly the same distance (.25 unit), to reach the center of each new Rectagon. The net result is, the center-to- center distance between adjacent Rectagons is the same as on the hex grid. Which was the goal. (Okay, for the wise guys out there, I know you could have a grid of "vertical" Rectagons, instead of the longer-distance-is-horizontal thing described here. So the height-to-width ratio is sqrt(3) to 1.5, which, geometrically speaking, doesn't count as a significant difference.)

Now I understand that the grid of squares on a viewscreen isn't going to perfectly accommodate the exact ratio of 1.5 to sqr(3), but it is much easier to draw a brick-wall grid of rectangles that are that are "close enough" to the ideal, than it is to draw a grid of hexagons. And therefore this can be a useful tool in the game-designer's toolbox.