[[ Most recent Update: 28 September 2012. ]]
Making a Dodecahedron, Part 1 (Layout, and making the first 4 cuts.)
The photo, above, shows a newly cut dodecahedron, fresh from the workbench, ready for sanding, varnishing, waxing, and polishing. In other words, it is about half way to being ready for prime time display.
Making a wooden dodecahedron is not a difficult task, but it is time-consuming, and requires careful attention to detail. I have never kept a detailed log of exactly how much time is required from start to finish, but I suspect it is at least a dozen hours, or so, if all the time for various tasks is included; tasks such as selecting wood, layout, cutting, sanding, varnishing, etc.
First, a quick review for those whose high school geometry has become a bit rusty (scroll down to ‘ Begin With a Cube ‘, below, if you don’t need a review).
There are five Platonic solids:
 The Hexahedron (three squares at each vertex)
 The Tetrahedron (three triangles at each vertex)
 The Octahedron (four triangles at each vertex)
 The Dodecahedron (three pentagons at each vertex)
 The Icosahedron (five triangles at each vertex)
Notice that the dodecahedron is the only Platonic solid that has pentagonal faces.
A dodecahedron has twelve faces that are congruent regular pentagons with three meeting at each vertex. Regular means that the sides of the pentagons are all the same length, and Congruent means that the are all the same size.
The ancient Greeks recognized that there are only five platonic solids. But why are there only five?
The key observation is that the interior angles meeting at each vertex of the Platonic solids add to less than 360 degrees.
 Hexahedron:Three squares at each vertex; 3×90 = 270
 Tetrahedron: Three triangles at each vertex; 3×60 = 180 degrees
 Octahedron: Four triangles at each vertex; 4×60 = 240 degrees
 Dodecahedron: Three pentagons at each vertex; 3×108 = 324 degrees
 Icosahedron: Five triangles at each vertex; 5×60 = 300 degrees
If you feel you need or want more detail regarding Platonic solids in general and dodecahedrons in particular, there are many fine websites that can provide more than you ever wanted to know. I think wikipedia is a good place to start, or simply Google the name, dodecahedron.
By the way, Plato was NOT the Greek who first recognized / discovered the “Platonic” solids – – one of those little quirks of history. History is, of course, that which gets recorded, but may or may not be what actually happened, and that’s a story for another day.
As with most any artifact, there are several ways to go about making a dodecahedron. For me, the most satisfying method is to cut a dodecahedron from a block of wood.
BEGIN WITH A CUBE
The ‘ block of wood ‘ is one of the platonic solids: the hexahedron, commonly called a cube.
There are a number of ways to go about trimming away the ‘ excess’ wood from the cube to reveal the dodecahedron within a cube. One method I have used successfully to build several dodecahedrons is detailed below.
The more accurate your cube, the more accurate your dodecahedron will be. Be sure your cube is square (all angles at the 8 corners are 90 degrees ).This will result in all the edges being exactly the same length. It is relatively unimportant what, exactly, the ‘ same length ‘ is.
The cube shown above measures about 8.2 cm on each edge, and was prepared from a ‘ 4×4 ‘ that was used as a post in an old redwood fence that I replaced several months ago.
I make most of my dodecahedrons from salvaged lumber, but any lumber yard worthy of the name can provide suitable lumber. One might expect new, store-bought lumber to be ‘ square ‘ and of uniform dimensions, but I have found this is not so. In any event, you will need to trim your cube to make it as close to perfect as possible. Stock lumber in the form of a 4×4 or 6×6 or 8×8 is a good place to start.
If money is no object, excellent wood for carving can be had from specialty suppliers.
 Divide each of the six faces of the cube into four squares.
You should use fine lines in order to increase the accuracy of your layout. I have found that a mechanical pencil using 0.5 mm lead works well for this purpose. Tiny errors accumulate during the twelve cuts you will be making, and this will cause your finished dodecahedron to be out of whack.
Take your time and get the layout lines as close to perfect as possible because these layout lines will be your guide for the twelve cuts required to turn a cube into a dodecahedron.
 Choose any one of the six square faces of your cube to be the ‘ master ‘ layout surface.
 Where the layout lines cross is the midpoint of this surface, point ‘ m ‘.
 Divide one-half of the Horizontal axis in half. This is point ‘ h ‘.
 The top of the vertical axis is point ‘ t ‘, as shown below.
 Place the point of your compass on point ‘ h ‘ and the pencil on point ‘ t ‘, then draw an arc that intercepts the other half of the segment at point ‘ i ‘.
 Use your compass to capture the distance from point ‘ i ‘ to the edge of the square at point ‘e ‘.
If your compass has a locking mechanism, lock your compass now to capture the length, i – e.
Call this captured length ‘ r ‘, for radius.
NOTE: Points h, t, i, and e have now served their purpose, so you can forget about them – – after first making sure you have captures the distance i – e for future reference.
Another NOTE: You will find the length of both the radius and the diameter of the circles to be essential for additional layout lines and for checking the accuracy of your cuts. More about this, later.
 Place the point of your compass at ‘ m ‘, and draw a circle with a radius of ‘ r ‘.
Congratulations! The diameter of this circle is the length of the edges of each of the twelve pentagonal surfaces on your dodecahedron. You can use this fact to check the perfection of your dodecahedron when you are finished, but don’t be concerned about that at this time. There are still many steps to complete before checking the accuracy of your finished work.
 Now, draw a circle on each of the remaining sides of your cube. TAKE CARE to insure the compass point is set EXACTLY where the horizontal and vertical lines cross. Tiny variations here will affect the following layout and cuts.
If your compass is not lockable, check the setting against the ‘ master ‘ surface before drawing each circle to insure that the radius is the same on all surfaces of the cube.
Remember to keep your layout lines as narrow as possible because they will determine the accuracy of your cuts.
 The edges on opposite sides of the cube will be parallel to each other.
 The edges on adjacent surfaces of the cube will be perpendicular to each other.
Additional layout lines, tangent to the circles, can now be marked on each of the six surfaces of the cube.
THE FIRST FOUR CUTS
Alert readers, such as yourself, will recognize that parts of these lines will be cut away. That is true
-BUT- . . . the parts that are not cut away will be useful for laying out further cuts.
OK, now we (YOU and I) can draw lines for our first four cuts.
NOTICE that the lines run from the center line if the circle on one surface to the circumference of the circle on the adjacent surface.
The shaded areas will be cut away from the cube.
My favorite tool for making the cuts is a Japanese style handsaw that cuts on the back- stroke. I strongly recommend that an electric-powered miter saw NOT be used to make the cuts, because I consider it to be far too dangerous when working with such small, blocky pieces.
Be that as it may, here are three views of what your former cube will look like after the first four cuts have been made and trimmed to match the layout lines.
In Part 2 of Making a Dodecahedron, we (You and I) will layout and do cuts five, six, seven, and eight.
- The Platonic Solids (geoffneilsen.wordpress.com)