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Dome Radius:3.00

Stru

tLength

Dom

e

Spher

e

A 0.759

30 60

B0.885

30 60C 0.883

60 120

D 0.938

70 120

E 0.974

30 60

F 0.895

30 60

4-way

connectors20 0

5-way

connectors 6 12

6-way

connectors65 150

The Theory

Platonic And

Archimedian

Solids

Face Variations

Projection

Variations

Truncations

Variations

Why Choose

Geodesic?

A geodesic is a line joining two points on the surface of the earth. A geodesic

structure is one that follows the surface geometry of a sphere. But there are many kinds

of dome structures that do this. What defines a dome as a geodesic structure? Geodesics

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Unlimited follows the lead of the Surrey University Space Structures Research Centre,

which, in our view, is the worlds leading authority on the configuration of space frame

structures, of which the geodesic dome is one. So here below are the main varieties of

domes, their names, and their defining features.

Schwedler Dome

This has ribs extending down from the crown

of the dome, rings extending horizontally

around the dome, and diagonals extending

from intersections between ribs and rings on

one horizontal ring to those on the next .

Ribbed Dome

This has ribs extending down from the crown

of the dome and rings extending horizontally

around the dome.

Lamella Dome

This has diagonals extending from the crown

down towards the equator of the dome, in

both clockwise and anti-clockwise directions,

and may or may not have horizontal rings,

but has no meridional ribs.

Diamatic Dome

This has what may be described as pie-

shaped sectors repeated radially around the

crown. Here, the apex of each sector has a

width of zero and, at its base, a sector is 360

degrees divided by the number of sectors .

Great Circle Dome

This is a dome not defined by the Space

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Structures Research Centre. We call it the

Great Circle Dome as all of its elements

follow more or less great circular paths over

the surface of the dome.

Geodesic Dome

This dome is rather different in its origins. It

is derived from one of the platonic or

Archimedean solids, or from a prism or anti-

prism - seePlatonic and Archimedean Solids,

and below.

Take for example the Icosahedron, which is a

platonic solid with 20 regular triangular

faces.

We take any face of this solid and subdivide

it in any way we choose, for example as in

the drawing on the left.

We then take this pattern, and project each

intersection in the pattern outwards onto the

surface of the sphere on which all the vertices

of the original icosahedron sit.

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We then replicate this around the whole of

the icosahedron, as left. We also have the

options of using a different base solid, a

different projection, a different face pattern,

and a different portion of the sphere. Thesepossibilities are explored in the following

pages

Platonic And Archimedian Solids

The Platonic and Archimedian solids (see the pictures below) are polyhedra, 3D shapes,

that are either completely regular in all respects, as are the platonic, or semi-regular, as

are the archimedian.

The platonic solids have regular length edges, are regular in size and shape of faces, and

have regular angles throughout. There are five of these.

The archimedian solids are variants of the platonic solids and have regular length edges,

but have two or three different faces. The angles are regular within a face, but vary from

face to face.

The choice between polyhedra as a starting point for a design depends upon the function

and desired aesthetics of the final structure. For example, a dome that wants to sit as a

conservatory in a corner between two wings of an existing building would be best as aquarter of a polyhedron with fourfold symmetry, such as the octahedron, but a free

standing greenhouse could be based purely on desired aesthetics and any of the

polyhedra could be used as a starting point.

The drawing below shows the five platonic polyhedra and how the archimedian

polyhedra are derived from them.

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Face Variations

Whichever polyhedron is the basis of our geodesic, we must decide upon a pattern to

map on to those of its faces we are going to use in our final structure. We may map any

pattern at all onto a face, and choose as many different patterns as there are faces, or

indeed map more than one pattern on to a face, or create layers of patterns on to faces to

create double or treble layer grid domes. Here we will use the icosahedron as the base

polyhedron and have a look at just a few possible face patterns we could choose.

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2 Frequency Triangulation

This is the simplest version, where we simply divide each edge of the triangle in two

and join each created mid-point to the next.

3 Frequency Triangulation

This is exactly as above except that we divide each edge into three.

6 Frequency Triangulation

Again a simple pattern, this time dividing each edge into six.

Projected 6 Frequency Pattern

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Here the pattern from the image immediately above has been projected to create the

geodesic triangle from the now patterned icosahedral one.

Geodesic projection of another single layer face pattern

This time a greater variation in final triangle size results from a variation in triangle sizein the pattern mapped on to the original icosahedral triangle.

Geodesic projection of a star onto a dodecahedral pentagon

This time a pentagon has been used since a star does not fit on to a triangular face. The

dodecahedron is a suitable base polyhedron for this.

Geodesic projection of another pattern mapped on to an icosahedral face.

Another more complex pattern is used here. The resulting dome is dimpled.

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A double layer of hexagons

Here we have a double layer grid that sits within a triangle.

Geodesic projection of above hexagonal double layer grid

The double layer grid now curves around the surface of our sphere.

Geodesic projection of above double layer grid created from the top five triangles of an

icosahedron

Projection Variations

Following our choice of base polyhedron, in this case the icosahedron, the portion of it,

in this case the upper half, and the face pattern, in this case a pattern of 16 triangles to

each face, we then have a variety of projection variations to choose from. Projectionrefers to the way in which our choice of now-patterned polyhedron is projected on to a

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surface, and also to the choice of surface. Here we shall be using projection outwards in

all directions away from a chosen point, but we will be looking at projection on to a

variety of different surfaces.

Projection on to a Sphere from the centre of the Icosahedron

Here our half icosahedron becomes a hemisphere and all our triangles are projected

evenly. The point of projection is the centre of the original icosahedron.

Projection on to a Sphere from above the centre of the Icosahedron

Here our half-icosahedron becomes more than a hemisphere, and the triangles are

projected unevenly, because the point of projection is now closer to the top of the

original icosahedron

Projection on to a Sphere from above the centre of the Icosahedron, and to one side

Here our dome again becomes more than a hemisphere and the triangles are all biased

towards the point of projection.

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Projection on to an Ellipsoid

Here the original sphere onto which the Icosahedron has been projected has beenelongated both horizontally and vertically. The projection is central.

Projection on to an Ellipsoid

This is another version of a central projection on to an ellipsoid

Projection onto a Paraboloid

This time the surface onto which we are projecting is a paraboloid.

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Projection onto an Elliptical Paraboloid

Now our surface is an elliptical paraboloid. This is again a central projection.

Projection onto a Hyperboic Paraboloid

Now our surface is a hyperbolic paraboloid.

Projection onto a Cone

Here our surface is a cone.

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Projection onto an Elliptical Cone

Now our surface is a cone that is stretched both vertically and horizontally.

Truncations Variations

Following our choice of base polyhedron, in this case the icosahedron, the portion of it,

in this case the upper half, and the face pattern, in this case a pattern of 16 triangles toeach face, we then have a variety of projection variations to choose from. Projection

refers to the way in which our choice of now-patterned polyhedron is projected on to a

surface, and also to the choice of surface. Here we shall be using projection outwards in

all directions away from a chosen point, but we will be looking at projection on to a

variety of different surfaces.

Full Sphere

Three Twelfths

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One Third

Five Twelfths

Hemisphere

Seven Twelfths

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Two Thirds

Nine Twelfths

Ten Twelfths

Why Choose Geodesic?

The Advantages of Using a Geodesic Design

1. Geodesics offer the strongest form of architecture known (because geodesic forms

follow a doubly curved surface and because the base is always a polyhedron which

already begins to approximate a sphere. The example of the strength of the shell of an

egg confirms the strength of a spherically based structure).

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2. A geodesic form creates the largest volume of space covered by the least amount of

material (the most efficient use of material to cover a given volume of space).

3. The largest unobstructed spans can be created using geodesic geometry.

4. Geodesic structures provide the most efficient form for the purposes of heating.

5. Spheres are aesthetically highly pleasing.

6. Spheres generate and embody potent energy fields.