CA1209315A - Geodesic dome - Google Patents

Abstract

ABSTRACT OF THE DISCLOSURE
A 3-frequency 3/8 sphere geodesic dome is modified by replacing one of the regular pentagonal sections having one edge adjacent the dome base with an irregular pentagonal section having a base edge longer than the base edge of the original regular pentagonal section, two side edges extending upwardly from the base edge, and two peak edges coinciding with two of the edges of the original regular pentagonal section.

Description

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WP:TTR9 GEODESIC DOME
This invention relates generally to geodesic domes, and has to do particularly with an improvement of what is known as a 3-frequency geodesic dome, in order to allow a larger size of entranceway than the conventional dome permits.
BACKGROUND OF THIS INVENTION
The design of most geodesic domes is derived from the icosahedron, one of the five platonic solids.
These polyhedra have unique properties, which have been studied since before the time of Plato. Of particular interest to the designer is the property that all vertices lie on the surface of a sphere. The icosahedron consists of twenty identical faces, each of which is an equilateral triangle. In some designs, small domes are often constructed directly from the icosahedron, with fifteen of the twenty faces being used to form the dome shell. Such structures are usually referred to as l-frequency or single-frequency domes. If each edge of a triangular face of the icosahedron is divided into two parts to form four triangles per icosahedron face, and if these smaller triangles are designed so that all six vertices lie on the surface of a sphere, the resulting structure is called a 2-frequency dome. When each edge of the triangular icosahedron face is divided into three parts, forming nine triangles per face, with the resultant vertices lying on a spherical surface, the result is the popular 3-frequency dome.
The height of a geodesic dome is generally described in terms of its diameter. 3-Frequency domes are typically designed to be either 3/8 (approximately), ox roughly 5/8 of the diameter in height. The 5/8 height is achieved by adding an extra row of panels.
GENERAL DESCRIPTION OF THIS INVENTION
The present invention provides an improvement in the conventional design of a 3-frequency 3/8 sphere ~!
geodesic dome by introducing a modification of the 3i5 length, position and associated angles of certain framing members, as well as a modification in the dome base, to make possible a ground level entrance approximately 60% larger than would otherwise be the case.
Early designs for domes of this type made use of a 15-sided base, with each side approximately equal in length. A modification later adopted by some dome builders replaced three contiguous triangular base panels with a single trapezoidal panel. This modification could occur a maximum five times around the base of the dome, reducing the number of sides from 15 to 10.
It was found that by utilizing the relatively large single trapezoidal panel rather than three smaller triangular panels, each in a different plane, the installation of doors and windows at ground level was considerably simplified. The trapezoidal panels were inclined inwards at approximately 11, and it was typical to frame up dormers from these panels to provide a vertical wall for the installation of doors and windows.
Even with the provision of this trapezoidal modification, however, a size limitation was encountered. Where a particularly large entrance was required for industrial or commercial use (as opposed to residential use), the trapezoidal panel had inherent limitations in terms of the height and width combinations of any rectangular opening. For example, in order to obtain an entrance wider than the trapezoidal panel top edge, it was necessary to reduce the height of the opening.
It was realized that more height could be obtained by building an entrance in one of the pentagonal sections which had an edge adjacent the dome base, but the width of this pentagonal section at its base is the same as the width of a trapezoidal panel at its top, and this established a limitation on the lateral width of any rectangular opening. It is therefore evident 3iS

that, in order to obtain an entrance both higher and wider than that permitted within a trapezoidal panel or within a pentagonal section, some further modification must be made to the dome shell, particularly to its base portion.
It is object of one aspect of this invention to provide a modification which will accomplish an increase in the permissible maximum dimensions of a rectangular opening in a 3-frequency dome, by introducing certain changes in the panel structure.
More particularly, this invention provides, in a 3-frequency 3/8 sphere geodesic dome having a base, the dome resulting from the truncation of an icosahedron and the division of the resulting regular pentagonal and hexagonal sections into groupings of 5 and 6 triangular faces respectively, a modification of the base portion thereof which comprises replacing one of the regular pentagonal sections having one edge adjacent the dome base with an irregular pentagonal section having a base edge longer than an edge of the original regular pentagonal section, two side edges extending upwardly from the base edge, and two peak edges coinciding with two of the edges of the original regular pentagonal section.
GENERAL DESCRIPTION OF THE DRAWINGS
One embodiment of this invention is illustrated in the accompanying drawings, in which like numerals denote like parts throughout the several views, and in which:
Figure 1 is a plan view of the base of an unmodified 3-frequency dome;
Figure 2 is an elevational view of a 3-frequency 3/8 sphere dome constructed on a 15 sided base;
Figure 3 shows a modification of the dome of Figure 2, again in elevation;
Figure 4 is a plan view of the base of the modified dome in Figure 3;
Figure 5 is a regular pentagon, representing the shape of one of the regular base pentagonal sections of ~2~ 5 the dome where the modification of this invention is applied;
Figure 6 is a plan view of an irregular pentagonal opening, representing the shape after modification;
Figure 7 is an elevation of the modified dome according to this invention;
Figure 8 is a plan view of the base of the modified dome of Figure 7, showing the alteration of the base as a result of the alteration of the pentagon;
Figure 9 is a plan view of a triangular face, identifying the apex angles;
Figure 10 is a schematic view to represent a dihedral angle;
Figure 11 is a partial vertical sectional view through a modified 3-frequency dome in accordance with this invention;
Figure 12 is a schematic view useful in the discussion of the geometric configuration of the modified dome; and Figures 13 and 14 are further schematic views useful to the discussion of the geometric configuration.
DETAILED DESCRIPTION OF THE DRAWINGS
Figures 1 and 2 are illustrative of the prior art 3-frequency 3/8 sphere geodesic dome built on a 15 sided regular polygonal base.
Figures 3 and 4 illustrate a known modification of the dome shown in Figures 1 and 2, according to which three contiguous lower triangles are replaced by a single trapezoidal panel, this being done at all five available locations in the case of the configuration shown in Figures 3 and 4. It will be understood that this conventional modification may be at one or more of the available locations.
Turning specifically to Figure 3, it will be seen that between the two trapezoidal panels 12 and 14 is located a regular pentagonal section 16 comprising five triangular faces 17, 18, 19~ 20 and 21. In this disclosure and in the appended claims, the word "section", where preceded by the word "pentagonal" or the word "hexagonal", refers to one of the composite surface features of a dome of the kind under discussion, for example the pentagonal shape shown at 16 in Figure 3, comprising triangles 17-21, inclusive.
S As another example, in Figure 3 a hexagonal section can be seen at 23, comprising triangular panels 25, 26, 27, 28, 29 and 30.
Elsewhere in the disclosure, the word "section" is sometimes used in its normal sense, to refer to a cross-section or similar drawing view.
The regular pentagonal section identified at 16 in Figure 3 is the location at which the modification of this invention is carried out.
Figure 5 is a geometric representation of a regular pentagon, corresponding to the shape of the pentagonal section 16 in Figure 3 representing prior art.
Figure 6 is a geometric representation of an irregular pentagonal shape, corresponding to the irregular pentagonal opening after the modification provided herein.
Attention is now directed to Figure 7, which illustrates the dome with the modification in question.
In Figure 7 the irregular pentagonal shape has a hypothetical base edge 35l two side edges 36 and 37 substantially perpendicular to the base edge 35, and two peak edges 40 and 41 coinciding with the uppex two original edges of the original regular pentagonal section.
In the drawing of Figure 7, the dome is constructed upon the polygonal base 43, the geometric shape of which will be discussed subsequently. Due to the height of the base 43, the actual entrance is higher than would be the case if the dome were constructed directly on grade.
A rectangular entrance 45 is limited at the top by a triangular panel 47 having the shape of an isosceles triangle, and is limited at right and left by narrow ~Z~lS

panels 50 and 52, respectively, each one having one longitudinal edge coinciding with the respective side edge 36, 37 of the irregular pentagonal section. The rectangular opening of the entrance continues downwardly below the true base plane 5~ of the dome, through the supporting base 43 and terminates at the grade level 5~.
Before describing the specific dimensions and angles of the panels contributing to the modification descxibed herein, it is useful to provide a brief explanation of construction methods and terminology for domes of this kind.
The framework of geodesic domes constructed from timber is typically designed using one of two distinct methods. One method involves cutting the framing members to a predetermined length and then connecting the ends with some form of hardware - referred to as 'hubs' - to create the framework or 'skeleton' of the dome shell. Some form of cladding, normally plywood, is then installed on the framing to form the dome shell. This method makes it possible for the builder to do all construction at the site, and is often used by the non-professional builder constructing a dome for his own use.
A second method, which seems to be widely used by dome builders, makes use of 'panelized construction'.
With this method, all components for the dome shell are precut, and each panel is fully assembled and pre-drilled for bolting to adjacent panels for assembly at the building site. This method enables the builder to manufacture the dome shell in a shop environment, and permits much faster assembly of the dome at the building site. With the panelized method of construction, however, strict attention must be paid to associated angles as well as lengths of framing members. The angles fall into two categories; usually called 'face' angles, and 'di~edral' angles. Face angles are the angles formed when two adjacent struts or framing members of a given panel meet. These ~Z~

determine the shape of a panel; (see Figure 9).
Dihedral angles are those formed when two adjacent panels meet; see Figure 10 for an example of dihedral angles. Dihedral angles determine the shape of a dome shell, and make it possible for the assembled panels to S approximate the surface of a sphere~ unless modified, as in the basis of this application, to partially depart from the spherical shape.
Attention is now directed again to Figure 7, and specifically to the panels 47, 50 and 52. The triangular panel 47 is simply a truncation of a regular pentagon, with face angles at the base of 36 each, and a face angle of 108 at the apex.
In terms of a unit radius of the dome (a convention in which the distance from the spherical lS centre of the dome to each ape~ is arbitrarily assigned the length one or unity), the two equal sides of the triangular panel 47 have a length of 0.4035482, and the remaining side has the length 0.6529547.
In Figure 3, the dihedral angle formed by panels 18 and 28 can be shown to have a measure of 168.64.
With reference to Figure 7, if we consider panel 47 to be part of the base of an oblique pentagonal right pyramid (which in actuality is the shape taken by the original five panels in this section), the dihedral angle formed by one of the eq~al sides of panel 47 and the adjacent triangular panel can be shown to be 156.30. The dihedral angle formed by panel 47 and each of panels S0 and 52 is determined by finding the inclined angle for panel 47 as well as the angle of inclination for panels 50 and 52. Panel 47 can be shown to be inclined at 63.44 with respect to the horizontal, while the inclination for panels 50 and 52 can be shown to be 79.19. The dihedral angle between them is therefore 63.44 plus 90 plus (90 minus 79.19), or 164.25.
The face angles and remaining dihedral angles for panels 50 and 52 are determined by the modifications made to the dome base, and will be described .~

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subsequently. Also, the necessary modification of trapezoidal panels 54 and 56 will be dealt with at a later point.
It is now appropriate to deal with the alterations of the dome base which are entailed by the modification already described. The calculations which follow will be best understood by first explaining the basis for the calculations.
It has already been mentioned that the distance of each vertex on the dome surface from a vertical axis passing through the spherical centre of the dome can be stated in terms of unit radius. Additionally, midpoints of lines joining corresponding vertices, and which are parallel to the dome base, may also be given a value in terms of unit radius. Through an analysis of the dome shell, points A and B in Figure 7 can be shown to have a radius, with respect to the axis, of 0.8572377. Since the length of AB is 0.6529547, the 'axis radius' of midpoint C is 0.7926342.
Figure 11 illustrates a section of the dome shell which is at a right angle to the elevation shown in Figure 7. The relative position of panel 47 is indicated, as well as C, the midpoint of the base of panel 47. CD is the height of panels 50 and 52 as shown in Figure 7, and the angle CDK is the angle of inclination, being the same as that of a regular trapezoidal panel, i.e. 79.19. Since the height of panels 50 and 52 is the same as for a regular trapezoidal panel, CD can be shown to have length 0.349483. The axis radius of C (indicated by OC in Figure 11) has been shown to have length 0.7926342.
Therefore the axis radius of point D (KD in Figure 11) has length OC plus (cos 79.19) (0.349483); i.e.
0.9591951.
With reference to Figure 12, the difference DE
between KD and the axis radius of the midpoint E of the base MN of the original pentagonal section, that is:
axis radius KE, is the key measurement for completing the design modifications.

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In Figure 12, KH, KM, KN, KP are all equal to the base radius for this style of geodesic dome, and can be shown to have a measure of 0.98516. Also, MN by construction has a length of 0.4035482. Therefore KE
can be shown to have a length of 0.96428. It follows by subtraction that DE has length 0.10608.
With reference to Figure 13, the angle JMN can be shown to measure 144Ø MR is equal to DE and therefore perpendicular to JL. Thus, in the triangle JMR:
JM equals DE/cos 54; i.e. 0.18047 JM equals DE (tan 54), i.e. 0.14601.
Referring to Figure 12, as well as Figure 13, JL
equals MN plus twice the length of JR. This is: JL
equals 0.69557. Also HJ and LP each equal HM minus JM.
Since HM is the measure of the base of a regular trapezoidal panel; i.e. 0.8070964; HJ and LP both have a measure of 0.62663.
This factor of 0.62663 is the length of the base of the trapezoidal panels which have been modified;
i.e. panels 56 and 54 in Figure 7. Panel 56, which is simply the reverse pattern of panel 54, is shown in more detail in Figure 14. With reference to this figure, ABCD deines the modi~ied panel while ABED
shows the shape of the panel prior to modification. By construction, AB, BE, and AD are all of length 0.4035482. DE is twice this length, namely 0.8070964.
The height of the panel, AF, measures 0.349483. Since the original panel ABEF has the shape of a bisected regular hexagon, and since BG is perpendicular to DE;
DG has length 1.5 times AB; i~e. 0.6053223. Since DC
corresponds to HJ in Figure 12, the measure of DC is 0.62663. Therefore GC has a length of DC minus DG;
i.e. 0.021308. Therefore the face angle BCD is equal to tan (BG/GC); i.e. 86.51. By subtraction, the face angle ABC measures 93.49. BC, which forms the fourth side of the modified panel, has a measure of 0.350132.
With reference to Figure 7, the width of each of panels 50 and 52 is determined by the required width of 93i~:i the entrance. That is, the width at the base of each panel will be the length of JL (see Figure 12) minus the width of the required entrance all divided by two.
The face angles correspond to the adjacent face angles of panels 56 and 54 in Figure 7, while the face angles next ~o the entrance opening are each 90Ø
The remaining measurement to be found is the dihedral angle formed by panels 52 and 56, and 50 and 54, in Figure 7. The expression for required angle has been calculated using a formula developed by the applicant for this particular purpose, however the formula itself can be omitted for the sake of brevity.
The dihedral angle ~s represented by the e~pression:
2cos (tanIe/tanF~, where Ie is defined to be the angle of inclination of the common edge of the adjacent panels, while F is the corresponding face angle of each panel. Since the height of the panel is 0.349483, and the angle of inclination of the face of the panel is 79.19; the 'vertical' height equals: sin79.19 (0.349483); i.e. 0.343281. It follows that the angle of inclination of the edge is the arc sin of the vertical divided by the length of the edge; i.e. Ie equals sin (0.343281/0.350132); or 78.65. Since the face angle has been shown to be 86.51, the dihedral angle has a measure of 144.64.
With reference to Figure 8, the line b represents the position of one of the base edges prior to modification, while broken line y represents the position after modification. It will be seen that the line has in effect "shifted" by distance x.
While one embodiment of this invention has been illustrated in the accompanying drawings and has been described hereinabove, it will be evident to those skilled in the art that changes and modifications may be made therein without departing from the essence of this invention, as set forth in the appended claims.

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Claims (14)

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE
PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS
FOLLOWS:

1. In a 3-frequency 3/8 sphere geodesic dome having a base, the dome resulting from the truncation of an icosahedron and the division of the resulting regular pentagonal and hexagonal sections into groupings of 5 and 6 triangular faces respectively, a modification of the base portion thereof which comprises replacing one of the regular pentagonal sections having one edge adjacent the dome base with an irregular pentagonal section having a base edge longer than an edge of the original regular pentagonal section, two side edges extending upwardly from the base edge, and two peak edges coinciding with two of the edges of the original regular pentagonal section.

2. The invention claimed in claim 1, in which, on either side of the irregular pentagonal section, the three contiguous triangular faces adjacent the dome base are replaced by an irregular trapezoidal face of which one edge coincides with a side edge of said irregular pentagonal section.

3. The invention claimed in claim 1, in which an entranceway is built into said irregular pentagonal section.

4. The invention claimed in claim 2, in which an entranceway is built into said irregular pentagonal section.

5. The invention claimed in claim 4, in which the entranceway is a rectangular opening with one edge adjacent the dome base.

6. The invention claimed in claim 5, in which the opening is bordered at either lateral side by a narrow panel having one longitudinal edge coinciding with the respective side edge of the irregular pentagonal section, and is bordered along an upper edge by an edge of a triangular face of which the other two edges coincide with the peak edges of the irregular pentagonal section.

7. The invention claimed in claim 6, in which the two narrow panels form with respect to a horizontal plane an angle of substantially 79°.

8. A 3-frequency 3/8 sphere geodesic dome having a base, the dome resulting from the truncation of the vertices of an icosahedron and the division of the resulting regular pentagonal and hexagonal sections into groupings of 5 and 6 triangular faces respectively, the lower portion of the dome being modified by the replacement of one of the regular pentagonal sections having a lower edge adjacent the dome base with an irregular pentagonal section having a base edge longer than an edge of the original regular pentagonal section, two side edges extending upwardly from the base edge, and two peak edges coinciding with two of the edges of the original regular pentagonal section.

9. The invention claimed in claim 8, in which, on either side of the irregular pentagonal section, the three contiguous triangular faces adjacent the dome base are replaced by an irregular trapezoidal face of which one edge coincides with a side edge of said irregular pentagonal section.

10. The invention claimed in claim 8, in which an entranceway is built into said irregular pentagonal section.

11. The invention claimed in claim 9, in which an entranceway is built into said irregular pentagonal section.

12. The invention claimed in claim 11, in which the entranceway is a rectangular opening with one edge adjacent the dome base.

13. The invention claimed in claim 12, in which the opening is bordered at either lateral side by a narrow panel having one longitudinal edge coinciding with the respective side edge of the irregular pentagonal section, and is bordered along an upper edge by an edge of a triangular face of which the other two edges coincide with the peak edges of the irregular pentagonal section.

14. The invention claimed in claim 13, in which the two narrow panels form with respect to a horizontal plane an angle of substantially 79°.