Christopher Glass, Architect38
Chestnut StreetCamden, ME 04843-2210 USA
Having been raised in the shadow of a Gothic cathedral (Washington, D.C.), I had always had reservations about modern architecture's ability to relate to its human users. The geodesic dome seemed to epitomize the problem. Since the geometry of the dome is merely an attempt to follow the skin of a sphere with straight struts, it is and appears to be a purely abstract engineering solution. The other image that had appeal was the Lunar Lander, a piece of engineering devoid of overt aesthetic appeal but having a significance derived from its completeness in its environment - "a man's home is his capsule". How, I asked, could such a capsule or dome be seen to be a continuation of classical design without explicitly incorporating classical decorative detail? The Platonic solids, especially the cube and the dodecahedron, seemed to offer an alternative that connected to classical proportioning systems and to the idea that geometry could relate to human scale. Over the years I played with the shapes at various scales, designing a "drafting pod" module and a small meditation house - neither taken further than the drawing board. Then a Japanese magazine invited Philip Johnson to judge a contest for a new Glass House, and I used the contest as an occasion to design a one-bedroom house plan on the order of Johnson's in the module. The contest entry sank without a trace, but I have put the ideas together for this article. A Platonic solid is a convex polyhedron whose every face is
the same regular polygon (and, to be precise, such that the same
number of faces surround each vertex). There are only five; three
of them have equilateral triangles as faces, one has squares,
and one has regular pentagons ( They are called Platonic because they are described in Plato's
dialogue (actually more of a lecture) with the astronomer Timaeus,
in which four of the five solids are described in detail, and
the other is presumably referred to (but then rather conspicuously
ignored). Timaeus, with no interruptions from Socrates, describes
the tetrahedron, octahedron and icosahedron (the three made out
of triangles) and the cube (made out of squares) as the geometrical
building blocks of matter. He then says "there was yet a
fifth combination which God used in the delineation of the universe
with figures of animals" [ Timaeus goes on to assign by "probability" the four shapes to the four basic elements, using reasoning that strikes us as fanciful but presumably impressed his contemporaries as rational and therefore likely. The smallest of the shapes, using the fewest parts and having the sharpest external angles, is the tetrahedron. It must, he states, therefore correspond to fire, the most active and literally pyrotechnical of the elements. The octahedron, resembling the tetrahedron but more complex, must be air. He says the cube is the most stable of the shapes (which is not true, as Fuller would point out) and therefore must constitute earth, and that leaves the rounder, drop-like icosahedron, which must be water. He then spends a long time explaining how these combine in increasingly unlikely ways to produce the appearances we call the physical world. This is the same dialogue in which Timaeus tells the story of Atlantis, and there is a suggestion from this juxtaposition that both the stories have the quality of useful but preposterous legends that will do as provisional explanations of the world as we experience it.[2] It is significant that Socrates never asks any questions, probing or otherwise, as he does of his other talk-show guests. It's as if it is not worth the effort, since this is so obviously fanciful. Timaeus never really discusses relating the shapes mathematically.
He describes only the characters of the elements and their interactions.
On the other hand, Euclid, in the culminating Book 13 of his
My construction is a modified version of Ghyka's. Essential to Ghyka's construction are the relationships between pairs of solids. Of the five solids, two pairs of them are "duals", in which the centers of the faces of one form the vertices of the other. This is true of the cube and octahedron, and of the dodecahedron and icosahedron (the tetrahedron is its own dual). However, for the purposes of constructing a Fuller-like space frame it is not very helpful to have vertices at the centers of faces. It is much easier and structurally stronger to connect vertices at edges, forming new vertices of more complex shapes. Fortunately, dualities apply to edges as well: dual solids have the same number of edges. For example, the dodecahedron has twelve pentagonal faces, but each of the face's edges is common to two pentagons, so it has 12 * 5 / 2 = 30 edges; the icosahedron has twenty triangular sides, and so by the same reasoning it has 20 * 3 / 2 = 30 edges. The other essential relationships that can be found among the solids are those of the diagonals: the diagonals of a cube form the edges of a tetrahedron. The midpoints of those diagonals, which are the midpoints of the edges of the tetrahedron, form the vertices of the octahedron. Since the diagonal of a square is the square root of 2, the edges of the cube and its inscribed tetrahedron are in the ratio . The octahedron's edges can easily be seen to be half those of the tetrahedron. So among these three shapes there is a pretty simple relationship. (Scale is irrelevant when considering ratios of quantities, but for the sake of definiteness take the edge length of the cube to be 1.) The fun starts with the dodecahedron. As the tetrahedron is formed by diagonals of the cube, so is the cube formed by diagonals of the dodecahedron. And the ratio of the length of the diagonal of a pentagon to its side is the Golden Section, commonly denoted by f. Therefore the ratio of the side of the dodecahedron to the side of the largest inscribed cube - one that connects pairs of vertices on each face of the dodecahedron in a regular rhythm of alternation - is , about 1:1.618, which by the "magic" of f is equal to f - 1 , about .618. (The mathematics behind this is that f is the unique positive number that satisfies f -1 = 1/f .) All of this brings me to the starting point of the construction
of my Pythagopod. Each of the faces of a cube can been envisioned
as the base of a shape like a hipped roof. The "ridge"
of that roof is one of the edges of the dodecahedron, elevated
above the surface of the cube far enough that the four "hips"
of that roof are all the same length as the ridge. Each face
of the cube has the same roof structure, but each face is rotated
90 degrees from the adjoining ones. It turns out that the distance
between ridges on opposite sides is f.
So the easy way to construct the dodecahedron is to construct
three rectangles (the blue rectangles in Figure 3a) whose sides
are f and 1/f
, and connect the corners of the rectangles to the vertices of
the cube ( For the icosahedron the construction is simpler, but the relationship is more complex. The scale of the icosahedron is chosen so that its edges and those of the dodecahedron intersect at their midpoint, made possible in part by the fact that each solid has the same number of edges (30), as detailed above. In fact, when the midpoints of the edges are made to intersect, they turn out to be at right angles to each other. And since opposite edges of the icosahedron are the same distance apart as opposite edges of the dodecahedron, they are f apart in relation to the cube in the dodecahedron. When such an icosahedron is constructed, its edge length will
turn out to be 1 - that is, the same as the inscribed cube. Thus
it can be constructed analogously to the dodecahedron. First,
generate three rectangles whose sides are 1 and f
(the green rectangles in Figure 4a). Then place these
within the cube, as was done for the dodecahedron, but ignore
the cube and connect the twelve vertices, in order to form the
triangular faces of the icosahedron ( So far, then, a figure containing all five Platonic solids
is obtained, with side lengths related to each other either by
Ö2 or by f
( The final step is to create the "skin" of the shape,
which is given by the outermost extent of the faces of the dodecahedron
and the icosahedron ( In fact, returning to the Surrounding the cube is the skin. The windows, being of glass
- which is technically a liquid, are the analog of water, which
Timaeus assigned to the icosahedron. And the dodecahedron is
the shape that makes the transition and introduces the Golden
Section. It is the "quintessence", the fifth element
uniting them all, the element Timaeus dismisses as the shape
of the animal world, but which we might think of as the basis
for organic life. In the house it should be thought of as metal.
Timaeus regarded metal as the result of the operation of fire
on earth, causing it to flow like water, so it is a good physical
analog to the immaterial fifth element ( The design that follows from this is basically just a cube house. It is axially oriented, with the entrances being on the sides that have the horizontal edges, and the "exedras" of stair and fireplace (or television, depending on your theology) on the sides that have the vertical edges. I account for the asymmetry of the tetrahedron (in relation to the cube's axes) by choosing the handedness that works with access to the stair and the bathtub. In the center of the downstairs is a square table whose inverted pyramid base follows the edges of the octahedron. Upstairs is the square bed, above which is a pyramidal ceiling light which is the top point of the octahedron, thus forming a virtual Masonic pyramid like the one on the Great Seal of the United States. The bedroom is open to the spaces below on three sides; the bathroom is tucked into the area over the entrance. The horizontal edges of the octahedron are flush with the ceiling plane below, but there is a glazed slot in the floor that allows the light to appear above as well. The diagonal edges of the octahedron penetrate walls, cabinets, and floors by passing through tiny tubes. The base of the structure is a cross consisting of one edge of the dodecahedron and one of the icosahedron. They are, of course, in f proportion. A final word about the entrances is in order. I mentioned
earlier the imagery of the Lunar Lander (technically the Lunar
Excursion Module or LEM). In a similar way, the entrances of
the Pythagopod are designed as hatches that lower to open and
rise to close. The lifting cables are shown in the model, and
when the hatches lift, the stairs and railing fold along with
the hatch planes. So when secured, the shape is complete. When
open, it is vulnerable ( A 'three-dimensional' model of the Pythagopod is included
for further exploration of the design. (see the o2c model at
the top of the page). In addition, four "snapshots"
have been generated from this model ( As I stated at the outset, this exercise was an opportunity to explore the relationships among the five solids, and to play the game of fitting human functions into an abstract geometry. Whether the proportions generated by the presence of in the geometry make the form beautiful I leave to the beholder to decide. As Plato has Timaeus suggest,
[2] Alternatively, of course, it could be a hint that
the lost superior culture of Atlantis was the source of the knowledge
of the physical world which Timaeus proceeds to explain. [3] His reference is in the following footnote: "Campanus
of Novara states in a subtle verbal antithesis that the Golden
Section (
Timaeus. Benjamin
Jowett, trans. New York: Bollingen Foundation.Ghyka, Matila C. 1946. Euclid.
introductory architecture studio at Bowdoin College and is trying to cut back on professional work to spend more time playing with toys like the Pythagopod.
Copyright ©2002 Kim Williams top of
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