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Haxagonal Mosque in 15mm - Fun with Math

I want to write a Wiki entry on geometry for terrain makers as well as some tricks for foam core. I figure I better make something with the research I am doing. Ok, I know, geometry...yuck, but without some careful planning, this would be really painful project.

So, a hexagonal mosque. Scale, 15mm. Intended use, Flames of War/Tunisia circa 1943. I can't seem locate any real-world hexagonal mosques in Tunisia, but this is more about hexagons than theological/historical correctness.

I have a 12"x7 3/8" plate I'll use as a base. I want to front door of the mosque to be one a face, and the vertexes to space the 7" width of the plate. The mosque will be forward on the place, with a walled garden and a pair of minarets in the rear. The mosque will have two levels, a lower lever with terrace wide enough for a Flames of War medium base.

Conceptual sketch of hexagonal mosque

OK, I have a lower level (six sides), a terrace that should just fit inside the lower level. An inner set of walls, also six, will be smaller. I want this upper level removable, to access the interior. I haven't decided yet how much of the interior will be accessible or whether the terrace will be attached to the removable part or the outer wall.

Math refresher: a hexagon is twice as far vertex to vertex as one of it's sides. Here is a new-to-me math word: apothem. An apothem is the shortest length from the middle of face. The radius, for polygons, is the distance between the center and the vertex or corner.

Parts of a hexagon

So the important things to know are the sides are the same and the radius and the apothem is about 0.866 times shorten than the sides/radius.

For my outer wall, the vertex to vertex diameter will be seven inches. The radius (center to vertex) is 3.5in, so the sides are also 3.5in long as well. The apothem of the outer wall is 3.5in x 0.866 or 3.03 inches. The face-to-face width will be 6.06 inches.

My terrace needs to fit inside the wall, no rabbiting or other fancy joints on the interior of a hexagon. It will need a support of some kind. And it will need to be smaller that the wall's exterior be 3/16th of inch for the foamcore, and this is measured from the apothem, because that is where the thickness of the wall is 3/16ths inch.

Now, I could try to measure and cut a hexagon, with 120 degree angles and so forth, but I think I'd introduce ever worsening errors and make a pretty bad hexagon. Instead, I am going to make some hexagons using my handy Draw tool in my office suite. I use LibreOffice, and it has a hexagon shape right in the drawing tool. You can specify the size of the hexagon in inches or cm, except you have specify it by the its width (vertex-to-vertex distance) and its hight (face-to-face)

So how big is the terrace hexagon? Well, its face to face width is 6.06 - 3/8ths = 5.69inches -- the face to face width of the walls less two sheets of foam core. But the vertex-to-vertex distance is not 7" - 3/8ths because the foam is at an angle at the vertexes. If the apothem of out wall is 3.03 inches, then the apothem of the terrace is 3.03" - 3/16th or 2.84". Divide the apothem by 0.866 to get a side or a radius and you get 3.28 inches. The vertex to vertex distance -- and the width of the hexagon, is 6.57 inches.

Build List:
6 outer faces at 1.75x3.5"
1 terrace (haxagon 5.69" tall and 5.67" wide with an inner hole
6 Inner faces at 1.28"x1.72. These will extend down to ground level
6 Inner faces at 2"x1.72. These will up from the terrace level.

The hexagonal terrace was pretty easy to cut: create a template in LibreOffice or favorite graphics program and print it. Cut it out, tape it to foamcore and trim very carefully. I found using a metal straightedge allowed me to focus on nice, clean square cuts.

I also printed a template for the roof of the second level, again using the office drawing tool. I used the center waste from the terrase, trimming in to fit.

Using a Template to cut a hexagon

The sides start simple, 3.5" long by 1.75" tall. Next, however, I have to cut a 60 degree bevel. I don't have 60 degree cutting tool, so I needed to cut them by hand. I want to remove a 7/64ths strip of backing paper. (The 7/64ths comes from the 3/16th thickness and a bit of trigonometry.) Once this 7/16ths strip is removed, the foam is easy to cut, using the remaining paper as a guide. I am getting close to sixty degrees -- a bit of filler will be needed at the end, since working in 1/64ths is very fine work indeed, and it easy to introduce errors.

Cutting bevels in three steps

Still, after one night's work, I have the terrace and the roof and six sides and they are fitting together well enough.

One night's work. Two hexagons and six outer sides

Here is a better image of how to cut the bevels:

  1. Cut the upper layer of paper at 7/64ths of inch.

  2. peel away the paper

  3. cut the foam down to the paper -- the foam is easier to cut than the paper, if your knife is sharp but not brand new

Cutting bevels in three steps

With the interior walls, I tried a dry-fit. The upper hexagon is just a tiny bit bigger than I planned, but the end result is this free-standing fit without any glue. Nothing but friction.

 dry fitting the mosque: Careful planning, careful cutting, no glue

Now with the exterior sides:

Hexagonal Mosque  dry fitting

I'd originally planned to have the interior accessible, with the lower interior walls having arches cut out. However, I am now leaning towards the entire interior being removable, terrace and all.

All the geometry is paying off because it looks amazing together. Great precision Thumbs Up

Assembled and basic painting. For all my math and careful cuts, I needed to trim the terrace hexagon a bit to fit properly. Still, I am really happy with how very perfectly hexagonal everything turned out.

I feel I need to add some details, but I am not certain what is missing.

I thought I might add towers to in the rear corners, but my first build attempt was a failure. Also, the Kairouan mosque in Tunisia has a square tower instead of the thin spires more typical. I may yet add them if I feel it needs it, but I don't want the model to be any taller for storage reasons.

Hexagonal Mosque assembled, painted

The interior...
Interior view of hexagonal mosque

I used the Roof hafez tomb roof image taken by PentoCelo on WikiMedia. (Creative commons licenses require this attribution. http://en.wikipedia.org/wiki/Mosaic#mediaviewer/File:Roof_hafez_tomb.jpg). The original image had a hole or skylight that appears as a white dot in the center. I used a cloning tool in the gimp to copy a bit of pattern into the round white circle and fill in the sky light, so the pattern is solid through out.

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