Back in my Dungeons and Dragons-playing
days, I was never very focused as Dungeon Master -- I'd draw elaborate
maps of and make up an elaborate history for the fantasy world i'd lead
people through, but somehow, I never got a campaign to last beyond one
day. *sigh*
Another silly aspect of the game that I wasted a lot of energy on was
the dice. As you are probably aware, AD&D uses dice shaped
like all five of the regular polyhedra.
My favorite ones were the dodecahedra. I'd
spend a lot of time just looking at the things, turning them around in
my fingers looking at them from different angles. One day, it occurred
to me that since it's a regular polyhedron, each opposing
pair of edges has to be the same distance apart. Given two
such pairs, an orthographic projection of a dodecahedron, when viewed
at the correct angle, will fit into a square:
__-__
__-- --__
__-- --__
- \ / -
| \ / |
| \_______/ |
| / \ |
| / \ |
-__ / \ __-
--__ __--
--__ __--
-
Your browser may vary.
Given three such pairs, the dodecahedron itself has to fit into a cube.
Many years later, this turned into a quest to figure out a method to carve
a dodecahedron out of a cubical block of something (wood, plastic, whatever).
I will share this method with you now.
N.B.
Noether shows us
how to make
a dodecahedron the other way, building one
up from a cube.
Materials:
Let's label the faces A, A', B, B', C, and C', such that A' is opposite
A, B' opposite B, and C' oppposite C. In an oblique view:
__-__
__-- --__
__-- --__
__-- --__
-__ __-
| --__ B __-- |
| --__ __-- | A'
C' | --__ __-- | (hidden)
(hidden) | - |<----'
'----->| | |
| A | |
| | C |
| | |
-__ | __-
--__ | __--
--__ | __--
--__|__-- ^------ B' (hidden)
-
First of all, the only points of contact the cube has with the dodecahedron
are along the three pairs of opposing edges mentioned earlier.
We need to lay out these edges on the cube. On any one face,
there is one edge, precisely centered in both dimensions. In each
opposing pair, the two edges are parallel, edges in different pairs are
always skew to each other. In a 3-D view:
__-__
__-- --__
__-- --__
__-- __- --__
-__ _-- __-
| --__ B __-- |
| --__ __-- | A'
C' | --__ __-- | (hidden)
(hidden) | - |<----'
'----->| A | | |
| -__ | | |
| --_ | | C |
| | | |
-__ | __-
--__ | __--
--__ | __--
--__|__-- ^------ B' (hidden)
-
If the side of the cube is s, The length of each side, after
some calculations, turns out to be s/phi2, which is
about s*0.391965...
But you shouldn't use a calculator to do this. You need a compass
and a straight edge, and your cubical block.
_____________e_____________
| ' /'\ |
| ' / ' \ |
| ' / ' \ |
| ' / ' | |
| ' ' ` |
| ' / ' \ |
| ' ' . |
|_ _ _ |/_ _ _'_ _ _._ _ _ _|g
| d c /f |
| ' |
| ' |
| ' |
| ' |
| ' |
| ' |
|_____________|_____________|
Divide each side into quarters, as in the above diagram. Let's call
the point where the two axes meet c.
On one side, divide one-half of one of the segments in half. Let's label
the midpoint of that segment d. Let's also label one of
the two points where the other axis meets the edge of the cube e.
Place the point of your compass on point d and the pencil on e,
and draw an arc that intersects with the other side of the axis, at f.
(I'm sorry, it's really hard to draw curved arcs with ASCII art.)
Now, set your compass to capture the distance between f and g, the point where the horizontal line meets the edge of the cube.
Erase the extra markings around points d and f (in fact, forget about
d and fentirely)
and place the point of your compass on c and mark off a point with the f-g distance you just captured onto the horizontal line, at h. You've just found one of the vertices of your dodecahedron! Now, mark the f-g distance to a point h' on the other side of c. Draw a line
segment between the points. This segment will be one edge of the
dodecahedron.
______________e______________
| ' |
| ' |
| ' |
| ' |
| ' |
| ' |
| ' |
|_ _ _ _ _____________ _ _ _ _|
| h'\ c /h |
| ' |
| ' |
| ' |
| ' |
| ' |
| ' |
|______________|______________|
Now use your compass to transfer similar line segments to the other
five sides of the cube. Make sure that the segments on each pair
of opposing faces of the cube are parallel to each other, and skew to the
segments on the other four sides (as in the 3-D view).
The remaining diagrams will contain one view of our work from side
A, and one from side C:
After you have measured and transferred the six edges onto the faces
of the cube, you can lay out the rest of the lines needed to mark off the
material to carve away. All six faces are the same.
Draw two lines across the cube, each one perpendicular to the edge
segment, and through an end of the segment.
Then extend the segment until it too crosses the entire face of the
cube:
A'
_____________v_____________
| ' |
| ' |
| ' |
|_ _ _ _ _ _ _'_ _ _ _ _ _ _|
| | |
| | |
| | |
C>| | |<C'
| | |
| | |
| | |
|_ _ _ _ _ _ _|_ _ _ _ _ _ _|
| ' |
| B |
| (B' opposite) |
|_____________'_____________|
^
A
B B
_____________v_____________ _____________v_____________
| ' ' | | ' |
| ' ' | | ' |
| ' ' | | ' |
| ' ' | |_ _ _ _ _ _ _'_ _ _ _ _ _ _|
| ' ' | | | |
| ' ' | | | |
| ' ' | | | |
C>|_ _ _ _'___________'_ _ _ _|<C' A'>| | | <A
| ' ' | | | |
| ' A ' | | | |
| '(A' behind)' | | | |
| ' ' | |_ _ _ _ _ _ _|_ _ _ _ _ _ _|
| ' ' | | ' |
| ' ' | | C |
| ' ' | | (C' opposite) |
|_______'_____ _____'_______| |_____________'_____________|
^ ^
B' B'
You will notice that for each opposing pair of faces on the cube, there
are six parallel line segments on the other four faces connecting them.
We can draw lines connecting the endpoints of these six lines on our original
pair of faces, so that each face also has a squashed hexagon drawn on it:
A'
____ ________v________ ____
| ' ' ' |
| / ' \ |
| ' |
|_ / _ _ _ _ _'_ _ _ _ _ \ _|
| | |
| / | \ |
| B | |
C>|/ (B' opposite) \|<C'
|\ | /|
| | |
| \ | / |
|_ _ _ _ _ _ _|_ _ _ _ _ _ _|
| \ ' / |
| ' |
| \ ' / |
|____.________'________.____|
^
A
B B
_____________v_____________ ____ ________v________ ____
| ' __-- --__ ' | | ' ' ' |
| __-- --__ | | / ' \ |
| __-- ' ' --__ | | ' |
|- ' ' -| |_ / _ _ _ _ _'_ _ _ _ _ \ _|
| ' ' | | | |
| ' ' | | / | \ |
| ' ' | | | |
C>|_ _ _ _'___________'_ _ _ _|<C' A'>|/ | \|<A
| ' ' | |\ | /|
| ' A ' | | | |
| '(A' behind)' | | \ | / |
| ' ' | |_ _ _ _ _ _ _|_ _ _ _ _ _ _|
|-__ ' ' __-| | \ ' / |
| --__' '__-- | | C |
| --__ __-- | | \ (C' opposite) / |
|_______'___--_--___'_______| |____.________'________.____|
^ ^
B' B'
You will notice that each edge of the cube is associated with a triangular
prism now marked out on the cube. These are what we cut away.
First, let's trim one of the wedges connecting faces C and C':
A
____ ________v________ ____
| ' ' ' |
| / ' \ |
| ' |
|_ / _ _ _ _ _'_ _ _ _ _ \ _|
| | |
| / | \ |
| B | |
C>|/ (B' opposite) \|<C'
|\ | /|
| | |
| \ | / |
|_ _._ _ _ _ _|_ _ _ _ _._ _|
| g l i |
k| |m
| |
|_____h_______ ________j____|
^
A
B B
___g_________v_________i___ ____ ________v________ g,i,l
| l | | ' ' '
| | | / ' \
| | | ' A1
k| A1 |m |_ / _ _ _ _ _'_ _ _ _ _ \
| | | | k,m
| (A' behind) | | / | \
| | | |
C>|_ _ _ _h___________j_ _ _ _|<C' A'>|/ | \ h,j
| ' ' | |\ | /|
| ' A, bottom ' | | | |
| '(A' behind)' | | \ | / |
| ' ' | |_ _ _ _ _ _ _|_ _ _ _ _ _ _|<A bottom half
|-__ ' ' __-| | \ ' / |
| --__' '__-- | | C |
| --__ __-- | | \ (C' opposite) / |
|_______'___--_--___'_______| |____.________'________.____|
^ ^
B' B'
The markings on one-half of face A have been obliterated.
The original straight lines stop at points h and j. The
diagonal lines on face B have been partially cut away, at points
g
and i. Stop now and re-draw them on the new face A1 while
the information is still on the cube. Connect g with h,
and i with j. Connect k with l, and
l with m.
B
___g_________v_________i___
| ' __-l-__ ' |
| \ __-- --__ / |
| __-- --__ |
k|-- \ A1 / --|m
| |
| \ / |
| |
C>|_ _ _ _\___________/_ _ _ _|<C'
| h' 'j |
| ' A, bottom ' |
| '(A' behind)' |
| ' ' |
|--__ | | __--|
| --__ __-- |
| --__ __-- |
|_______'____-_-____'_______|
^
B'
Repeat this until all four of the wedges connecting C and C' have been
trimmed away:
B B
___ _________v_________ ___ ________v________
| ' __-- --__ ' | ' ' '
| \__-- --__/ | / ' \
| __-- --__ | ' A1
|- \ A1 / -| A2' / _ _ _ _ _'_ _ _ _ _ \
| (A2' behind) | |
| \ / | / | \
| | |
C>|_ _ _ _\___________/_ _ _ _|<C' / | \
| / \ | \ | /
| A2 | |
| / (A1' behind)\ | \ | /
| | _ _ _ _ _ _|_ _ _ _ _ _
|-__ / \ __-| A1' \ ' / A2
| --__ __-- | C
| / --__ __-- \ | \ (C' opposite) /
|___._______--_--_______.___| .________'________.
^ ^
B' B'
Now trim the wedges connecting B and B', remembering to re-draw
layout lines after each cut:
B B
_____________v_____________ ________v________
| ' __-- --__ ' | ' ' '
| \__-- --__ / | / ' \
| __-- --__ | ----____ ' ____---- A1
|- \ A1 / -| A2' / --'-- \
| (A2' | |
| \ behind) / | / | \
| C1 C2' | C2 | C1
| \___________/ | / (C1' | (C2' \
| / \ (C1' | \ beh- | beh- /
| (C2 A2 beh- | ind) | ind)
| beh- / (A1' \ ind) | \ | /
| ind) behind) | |
|-__ / \ __-| A1' \ ____--'--____ / A2
| --__ __-- | ---- ' ----
| / --__ __-- \ | \ ' /
|___._______--_--_______.___| .________'________.
^ ^
B' B'
Finally, trim away the last four wedges to reveal the dodecahedron:
_ _________________
B1 __-- --__ B2 ' '
__-- --__ / B2 (B1 behind) \
__-- --__ ----____ ____---- A1
- \ A1 / - A2' / -- -- \
| (A2' | |
| \ behind) / | / | \
| C1 C2 | C2' | C1
| \___________/ | / (C1' | (C2 \
| / \ (C1' | \ beh- | beh- /
| (C2' A2 beh- | ind) | ind)
| beh- / (A1' \ ind) | \ | /
| ind) behind) | |
-__ / \ __- A1' \ ____-- --____ / A2
--__ __-- ---- ----
--__ __-- \ B2' (B1' behind) /
B2' --_-- B1' ._________________.