1. 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. Using a compass, draw a circle.
2. Draw a horizontal line through the center of this circle.
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.

2. 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.
3. 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.
4. 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.

1. 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.
2. 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.

1. 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.
2. 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.

3. 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.
4. 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.

1. . 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.
2. . 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.

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 four- frequency 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 tools, there's more fun ahead.

• 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.

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.