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```How to Draft a Geodesic Dome
Spencer Hunter, 1996
(Permission granted to distribute freely in unaltered form.)
Links to public-domain images that accompany this document may
be found one level back in my Comprehensive Anticipatory Design
Science directory, or at the URL:
Abstract: one method of drafting a geodesic dome, known as the
alternate breakdown, is to subdivide one face of an icosahedron
and project the derived points out to its circumscribing sphere.
The points on the sphere are then connected by lines, and
auxiliary views and rotations can then be performed to obtain the
desired true size of each of the triangular surfaces, giving face
angles and chord lengths. Further rotations and auxiliary views
can be produced as needed to get axial and dihedral angles, but
those won't be covered here.
If the drawing is executed carefully enough, an actual model can
be built from it, which I have successfully done with index cards
and glue. Bear in mind, though, that this is an analog technique
and is therefore imprecise. For large-scale construction, either
use a computer aided design program such as AutoCAD to derive
precise values, or refer to published tables of chord factors
such as those by Joseph Clinton in Domebook 2.
I. How to draft the needed portion of an icosahedron in two
views.
An icosahedron is one of the five basic Platonic solids
consisting of twelve vertices (twelve points, surprizingly, are
the most equidistant points one can cram onto the surface of a
sphere) and twenty equilateral triangular faces. It helps to
know that surrounding each vertex are five equilateral triangles.
A. The top or plan view of the topmost five triangles of an
icosahedron with its apex facing straight up, then, would be
a perfect pentagon. The sides of the pentagon would
represent the true length of each edge of the icosahedron,
and the edges leading up to the apex would appear as spokes
in this pentagon to its center.
So, the first step in drafting the geodesic dome is to
construct this pentagon with its spokes.
1. Draw a horizontal line.
2. Using a compass, draw a circle with its center on
the line.
3. Bisect the distance from the left intersection of
the circle and the center. Label this point m.
4. Construct a perpendicular bisecting line to the
horizontal line through the center of the circle (not
point m). Label the intersection of the new line with
the top of the circle point T.
5. With the compass point on m, set the span to the
length from m to T and swing an arc down to the
horizontal line. Label the intersection point p.
6. With the compass point on T, set the span to the
length from T to p. This is the length of one side of
the pentagon. Use the compass to mark off this length
about the circumference of the circle. Connect these
new points with straight lines to form a pentagon.
7. Draw straight lines from each of the vertices of the
pentagon to the center to form the spokes.
The top point of this pentagon is T; label the center A,O
(for "Apex" of the icosahedron, and "Origin" or center of
its circumscribing sphere that lies directly beneath the
apex in the plan view); and the two points that form the
base line at the bottom of the pentagon, from left to right,
B and C respectively.
B. Next, draw in a folding line to the right of the pentagon
parallel to line TA. This is where the side elevation view
will be constructed. Draw projection lines to the right
from points T; A,O; and line BC perpendicular to the folding
line.
C. Along the projection line from A,O in the side elevation
view, arbitrarily select point A that is the apex of the
icosahedron. Set your compass span to the length of one of
the sides of the pentagon in the plan view, then with the
compass point on A in the side elevation, draw an arc that
intersects the projection line from T away from the folding
line. This intersection is point T in the side elevation.
Draw in line TA that is shown in true length in the side
elevation view.
D. In the side elevation view, draw a line perpendicular to
the projection line at point T straight down until it
intersects the projection line from B and C. Label this
point C, B (point B is hiding behind point C in this view)
that represents the line BC seen straight on as a point in
the side elevation. Again in the side elevation, draw a
line from A to C, B that is the true length of the altitude
of equilateral triangle ABC and also represents triangle ABC
seen edge-on as a line in the side elevation view. This is
all of the icosahedron we need to draw in order to draft a
geodesic dome.
II. How to draft the needed portion of the circumscribing sphere.
A. In the side elevation view, construct a perpendicular
bisecting line to line AT. Where this line intersects the
projection line from point A, O in the plan view is point O
in the side elevation.
B. With the compass point on point O in the side elevation,
set the span to the length from O to A, the apex of the
icosahedron, which is the same length as O to T. Swing an
arc from A down past point C, B. This is all of the
circumscribing sphere we need to draw in order to draft a
geodesic dome.
III. Subdividing one face of the icosahedron.
A. Now it's time to arbitrarily decide on the degree of
breakdown, or frequency, of each equilateral face of the
icosahedron, which will be projected out to the surface of
its circumscribing sphere to form a dome. If you decide on
a frequency of one, we're done! You have a geodesic dome
consisting of five equilateral triangles (yawn). A
frequency of two means to find the midpoints of the edges of
each face and connect them to subdivide the face into four
smaller equilateral triangles. A frequency of "v" means to
divide each edge of all the faces into "v" parts and connect
all the resultant points in a triangular grid that
subdivides each face into "v"-squared number of smaller
equilateral triangles.
B. For this document, we will set "v" equal to four, since
that frequency may be the smallest breakdown we can reliably
get away with using this method of construction; because it
produces a very aesthetically pleasing dome; and perhaps
because it's also the frequency Buckminster Fuller chose for
his first large-scale commercial dome commissioned by the
Ford Motor Company. Four is an even-frequency breakdown,
and all even-frequency breakdowns using icosahedra allow us
to construct hemispherical domes without the need to
truncate any of the surfaces.
In the plan view, subdivide each edge of triangle ABC into
four equal parts. Connect all the resultant points with
straight lines to form a triangular grid that should divide
triangle ABC into sixteen smaller triangles.
C. Label all the new points, including those formed by line
intersections in the triangular grid. Starting in the plan
view from point A, move down one level on the grid in
triangle ABC and label the first two new points a1 and a2,
left to right, respectively. Move down to the next level
and label the next three new points b1, b2, and b3, left to
right, respectively. Move down to the next level and label
the next four new points c1, c2, c3, and c4, left to right,
respectively. Finally, move to the bottom level of triangle
ABC that is line BC, and label all five points d1, d2, d3,
d4, and d5, left to right, respectively. Note that d1 is an
alternate label for point B, and d5 is an alternate label
for point C. d2, d3, and d4 are the new points created
along with the triangular grid.
D. Project all the new points into the side elevation view.
a1 and a2 project onto one point along line AB in the side
elevation, and that point is labeled a2, a1 (since a1 lies
behind a2 in this view). Similarly, b1, b2, and b3 project
onto one point along line AB in the side elevation and
labeled, of course, as b3, b2, b1. c1, c2, c3, and c4
project onto one point along line AB that's labeled c4, c3,
c2, c1, and the points d1, d2, d3, d4,and d5 project onto
one point at the end of line AB labeled d5, d4, d3, d2, d1
(that is an alternate label for point C, B).
If you had chosen a higher frequency, you would need to
label and project more points; had a lower frequency been
chosen, fewer points would be labeled and projected.
IV. Projecting the derived points out to the circumscribing
sphere.
Projecting in this sense means radially out from the center point
O of the circumscribing sphere, not the usual orthographic
projection between the plan and side elevation views with respect
to the folding line.
A. Note that the projection lines from point O in the plan
view to points b2 and d3 are parallel to the folding line.
Therefore, these lines will appear as true length and, more
importantly, as true slope in the side elevation. We can
then draw projection lines out from point O to points b2 and
d3, respectively, in the side elevation view and extend them
to find out where they intersect the circumscribing sphere.
Label these new points b2' and d3', respectively, that
truly lie on the circumscribing sphere. Project
(orthographically) b2' and d3' back into the plan view,
where they lie along the radial projection line from O to b2
and d3 in the plan view.
B. The radial projection lines from O to most points in the
triangular grid of triangle ABC in the plan view, however,
are not parallel to the folding line, so we cannot reliably
extend them to the circumscribing sphere shown in the side
elevation to obtain where they would project.
The solution to this problem is to rotate any given radial
projection line in the plan view so that it _is_ parallel to
the folding line, determine where it intersects the
circumscribing sphere in the side elevation, then rotate it
back to its original position with the derived projected
point shown.
Let's use point c2 as an example. In the plan view, extend
a radial projection line from O to and slightly beyond c2.
Set the compass point on O and adjust the span to the length
from O to c2. Swing an arc from c2 over to the extension of
line TA and label that intersection rc2 so that line Orc2 is
now parallel to the folding line. Orthographically project
rc2 into the side elevation. Circular rotations in the plan
view translate into linear rotations in the side elevation
(a two-dimensional circle seen from the side is simply a
vertical line). To find rc2 in the side elevation, draw a
line down from c2 in the side elevation until it intersects
with the projection line from rc2 in the plan view. This
intersection represents point rc2 in the side elevation.
Next, extend a radial projection line from O out to rc2 in
the side elevation and extend the line until it intersects
the circumscribing sphere. Label this point rc2' and
orthographically project it back to the plan view where it
intersects the extension of line TA that represents rc2' in
the plan view. With the compass point on O in the plan
view, reset the span to the length of line Orc2' and swing
an arc back to the original radial projection line Oc2. The
intersection between that arc and the extension of line Oc2
is c2', the radial projection of c2 out to the
circumscribing sphere. In the side elevation, rotate rc2'
back to the original radial projection line that extends
from O to c2 and beyond by drawing a line up from rc2' until
it intersects the extension of line Oc2, that is c2' in the
side elevation. You can double-check the accuracy of your
drafting by seeing if c2' in the side elevation does indeed
line up c2' in the plan view.
All the other radially projected points a1', a2', b1', b3',
c1', c3', c4', d2', and d4' can be determined in both views
by this same process. d1' and d5' are the same as d1 and
d5, that are the same as B and C, two points of the
original icosahedron that lie on the circumscribing sphere.
V. Connecting the projected points.
This should be a fairly straightforward process in the plan view.
Simply connect with straight lines the projected points as the
points they were derived from in triangle ABC are connected.
In the side elevation, this is a bit more confusing, as a1' is
hiding behind a2' and hence triangle Aa1'a2' appears edge-on as a
line, as does triangle b2'a1'a2', triangle b2'c2'c3', and
triangle c2'c3'd3'. Also, only half of what appears in the plan
view can be seen in the side elevation, with the other half
consisting of invisible lines that hide behind visible lines.
However, when all the points are properly connected, you now have
a two-view three-dimensional rendition of 1/20th of a fourfrequency geodesic sphere. I believe computer aided design
programs would now give you precise values for all lengths and
angles in the drawing, but if all one has are standard drafting
VI. Deriving usable templates from the drawing.
Fortunately, only five triangular templates are needed to
construct the entire 320-triangle geodesic sphere, or a beautiful
160-triangle hemispherical dome. To avoid any further confusion
(I hope), allow me to specify those five triangles with single
letter descriptions:
triangle Aa1'a2' => triangle A
triangle a1'a2'b2' => triangle B
triangle a2'b2'b3' => triangle C
triangle b2'c2'c3' => triangle D
triangle c2'c3'd3' => triangle E
In standard applied descriptive geometry, one could construct
folding lines parallel to the edges of each of these triangles to
determine their true lengths in auxiliary views, and then
construct the templates on a separate sheet of paper, but there
is a much simpler way. Notice that triangles A, B, D, and E
appear edge-on in the side elevation. If we rotate those
triangles so that they are parallel to the folding line and
project them back into the plan view, they will appear as true
size in the plan view, with all face angles and edge lengths
shown in proper proportion.
Triangle C is not shown edge-on, so we need an auxiliary view
that does show it so. Draw a construction line across triangle
C in the side elevation parallel to the folding line and project
it back into the plan view, where it is shown in true length.
Create a new folding line perpendicular to this construction line
in the plan view and project triangle C into the new auxiliary
view, where it should appear edge-on (since the new auxiliary
view is related to the side elevation view, the distances from
the new folding line to all points will be the same as those from
the folding line to all points in the side elevation). In the
new view, rotate triangle C until it is parallel to the new
folding line, and project it back into the plan view where it
will now be shown in true size.
Usable templates can now be created from the drawing by simply
copying all five true-size triangles onto a separate sheet of
paper.
VII. Constructing a hemispherical model.
You will need thirty copies of triangle A, thirty copies of
triangle B, thirty copies of triangle E, ten copies of triangle
D, and sixty copies of triangle C. Triangle C translates to two
different mirror-image triangles in the model; construct thirty
copies, then flip the template around to create the other thirty
mirror-image copies if needed. If a higher frequency was chosen,
there would be more of these mirror-image triangles to contend
with.
Careful labeling of all triangles will allow successful
construction of the model. Constantly refer back to your drawing
to see how they all fit together.
From <http://www.u.arizona.edu/~shunter/howto.txt>
```