Content uploaded by Christopher Kitrick

Author content

All content in this area was uploaded by Christopher Kitrick on May 10, 2015

Content may be subject to copyright.

Unified Approach to Class I, II & ru Geodesic Domes

Christopher J. Kitrick

Reprinted from

INTERNAIIONAL JOURNAL OF

SPACE, STRI.ICTTIRE,S

Volume 5 Nos. 3 &,4 1990

MUTTI-SCIENCE PTTBLISHING CO. TJTD.

107 HIGH STREET, BRENTWOOD, ESSEX CMI4 4RX LINITED KINGDOM

A Unified Approach to Glass l,ll &

lll Geodesic Domes

Christopher J. Kitrick

Wavefront Technologies, Inc., 530 E. Montecito St., Santa Barbara, California. 93103, USA

(Received 31st May 1989; revised version received l9th Rbruary 1990).

ABSTRACT: This paper outlines a unified approach for generating all three

classes of geodesic domes. The approach, which allows one method to generate

configurations for all three classes, is presented in detail along with ten specilic

methods from which numerous actual configurations can be generated. The new

approach is especially useful in generating the more difficult class III domes.

Also a number of graphs are included to illustrate each of the given ten methods'

relationship of frequency to the number of different edges, and faces, which are

important design criteria. Lasfly, d number of older, independently developed

methods, which are encompassed by the new approach, are cross-referenced.

1 . Geodesic Glassification

Current geodesic classification is the result of add-

ing two subscripts (D,c) to the Coxeter notation for

polyhedra {p,ql. Thus for the icosahedron the

notation reads {3,51u,r''',' . The values of b and c fatl

into three categories:

Class I b>0 c:0 (Fig. 1),

Class II b>0 c-b (Fig. 2),

Class III b>0 c)0 and c*b (Fig. 3).

Class III configurations are enantiomorphic;

exhibiting right and left handedness when the

values of b and c are reversed. In this paper all class

III (b,c) pairs will have b greater tharl c.

Most geodesic domes constnrcted are either

class I or II, due to the availability of methods

which generate tessellations with a linear increase

in complexity. Class I are the easiest to use

especially when near equatorial truncation is

desired. Class III have no easy tnrncation arcs,

making them require the most attention when a

tnrncation is desired.

Intemational Joumal of Space Structures Vol. 5 Nns'. 3&4 1990

Clas s I

Figure I

Class I I

Figure 2

Class I I I

Figure 3

lli

(b.c) - ({,0)

(b.c) - (2,21

tu.cl - (2.[ )

The number of vertices, edges, and faces for any

(b,c) pair is a function of r where:

t: b2 + bc + c2

2. The lcosahedron

The icosahedron is the typical base geometry used

for geodesic domes because its Schwarz triangle

(U6 of an icosa face) is the smallest single (enall-

tiomorphic) sphere section (L/120) that can be

repeated to cover the entire spherical surface (Fig.

4). The Schwarz triangle will herein be referred to

as the lcd (lowest common denominator) triangle.

Atl geodesic vertices on an icosahedron can be des-

cribed by their relationship to the lcd triangle. All

the concepts presented herein are also applicable

to the Schwarz triangles of both the regular

octahedron and tetrahedron.

Schw arz Triangle

l/6Icosa Face

Fig. 4.

The relationship of t to the number of vertices,

edges, and faces for the icosahedron is:

vertices

edges

faces

3. Methodology

+2,

The methodology presented in this paper is

designed to include all classes and frequencies (b,c

pairs). It is not intended to optimize any par-

ticular one.45

224

Unified Approach to Geodesic Domes

Grid subdivision of the lcd triangle

f = 12

Fig. 5.

Class I

Fig. 6.

Class I I

Fig. 7.

International Journal of Space Stntctures Vol. 5 Nos. 3&4 1990

: 10r

: 20t,

- 30r.

C.J. Kitick

The basic approach involves the modular sub-

division of the lcd triangle into a rectangular grid

(Fig. 5). The number of divisions along an edge of

the lcd triangle is denoted by frequency A.For

every (b,c) pair there is a corresponding frequency,

such that all vertices of the tessellation will falI on a

grid intersection. For simplicity we will consider

the planar lcd as it appears on the face of the

icosahedron as a (30", 60o, 90o) triangle where all

rectangular cells are the same size with propor-

tions of /3/3 to 1.

For Class I tessellations frequency equals b/2

and each triangle is two cells wide and three high

(Fig. 6). Class II tessellations have a frequency

equal to b and each triangle is one cell wide and

two high (Fig. 7).

Class III Ske\M Ratios

Fig. 8.

Class III tessellations fall at a skew angle to the

grid. The difficulty is in finding the frequency so

that all vertices fall on grid intersections. Fig. 8

illustrates the basic situation of class III. The

givens are b,c, r:l and the angle of I20'. The

following notation describes the procedure used to

determine the correct frequency (fl for any class III

(b,c) pair as well as the individual offsets (m, n, ffib,

rltg, ftb, nr) for a single triangle (Fig. 9).

International Journal of Space Stntctures Vol. 5 Nos. 3&4 1990

Skew Ratios for (3,5)2,1

.k Eb:4 \

Fig. 9.

ob=5

fla:3

d

cosB

cosC

sin^B

sinC

dx

dy

m'

n'

m

n

tt

a

mb

mc

nb

nc

f

J)

is:

Using the above terminology (m, n, ffib, tlts, N6, ns,

for classes I dnd II and a few examples of III

Table 1

Class (b,c) m n.,, m6

I (any)2 3 I

U (any)l 2 I

m (2,1)5 9 4

m (3,1) 7 t2 5

m (3,2) 8 15 7

m (4,1)3 5 2

m 9,2) 5 9 4

m (4,3) 1l 2r lo

mcn6ncl

230

lll

563

7e3

896

341

563

1l 12 9

b/2

b

7

l3

19

7

t4

37

(Fig.5)

(Fig.7)

":'""';""""

a-, I

."' ," \

att

.t' ,i ia

aaa

." ,' ta

.aI

.t' ,' ta

aat

,t-,t

aa

a,

."

aa

aa

aa

a

.a

.,

aa

.t

.,

aa

a

.t

.,

.,

,a

.a

.a

.,

.a

aa

.,

.a

tl

a

a

a

m.:5

= y'@, + c2/4 + bc + t/qcT,

= (b/Z + c)/d,

: (c/2 + b)/d,

= (\/3b)/(u),

= (t/tt)t(?d),

= cosc - cosB,

= tfi/ux,

= cosC(b-c)lax : Note: b)c,

= (sirrB + sinO(b-c)/dy : Note: b)c,

: m'/g[eatest common multtple (!fr', ft'),

= n'/gyeatnst common multiple (m', fi'),

= cosC/m,

= t/ltl F,

= cosB/p,

: cosC/p,

= sinB/d,

= sinC/d,

= (bm, + cm/Z = d/(Ztt).

225

Unified Approach to Geodesic Domes

Class I I I

(b,c) - (2,1)

All grid intersections are defined by a coor-

dinate pair denoted by (ij) where:

i+j<f

Each method described in this paper provides a

unique one to one mapping of (U) pairs to (x,y,z)

coordinates. Thus for a particular geodesic

tessellation (b,c) the icd triangle is divided to the

correct frequency (f) andthe appropriate (ii) coor-

dinates are located for all unique vertices. Next

using a specific method the (ii) coordinates are

mappecl to actual (x,y,z) from which edge and faces

can be calculated. For example {3,5}r,, (f:7) has

only two unique (ij) coordinates (7,0) and (2,3)

(Fig. 10). All other vertices can be determined from

simple transformations.

226

4. The Mappings

This section describes in detail the mapping

calculations necessary for mapping (ii) coor-

dinates into (x,y,z) coordinates.

4. 1 Nine Methods for a Right Spherical

Triangle

The spherical methods operate on the spherical

lcd triangle (Fig. 11). Notation used for all nine

spherical methods is:

a - opposite arc side,

b - adjacent arc side,

c - hypotenuse arc side,

A - angle opposite arc side a,

B - angle opposite arc side b,

International Journal of Space Structures Vol. 5 Nos'. 3&4 1990

C.J. Kirick

(ij) integer coordinate of a point on

the lcd grid,

"f - frequency of grid,

(x'yo),r,, angular equivalent of (U): sub-

script denotes side,

(x,y,z) 3d coordinates of (ij) and

(xo, /o)u*ir.

It must be noted that b andc here in the context of

these methods bears no relationship to the

{p,qI u,,.

Sphe rLcal lcd Triangle

dinatcs (x",! )rr.rzr and thcn convert to (x,y,z) (Fig.

t2).

4.1.1 Method ua

Side a is subdivided by J' into A,a arc segments. At

each La distance a perpendicular arc is projected

until it intersects side c. All grid intersections lie on

these projected arcs. Grid points with the same i

value are the same arc distance away from side a.

For any (ij) position the (x'.)r"),.* is (Fig. 13).

La - o/f,

xo - arctan (sin(iA a)tan B),

y" - jLa,

axis - y.

4.1.2 Method ab

Side a is subdivided by -f into La arc segments.

From side c arcs are dropped to be perpendicular

to side b such that the opposite sides of these

smallertriangles are multiples of La.All grid inter-

sections lie on the arcs perpendicular to side b.

cos(y")

cos ( y' )

A=

B=

a

b=

JO

60'

90'

20.9C5'

31.717"

37.377"

asin (sinAsinb/ sinB)

alanQ) /2

asin (sinb/sinB)

The naming convention for the methods is

derived from the following

Metho d aa(b)

The Iirst letter denotes which side of the lcd is

uniformly divided by .f.The second letter denotes

the first side used to project or drop perpendicular

arcs. The optional third letter denotes the last side

used to project or drop perpendicular arcs.

The convention for Figs. 13-21 is that solid lines

represent continuous arcs.

The procedure for finding (x,y,r) from (ij) is to

first determine the intermediate spherical coor-

International Journal of Space Structures Vol. 5 Nos'. 3&4 1990

X=

Y=

L

sln

sin

COS

xn

Y,

xn

srn

sin

COS

xo

Y"

xo

X=

Y=

Lcos ( x'

cos ( y'

211

aXlS : XaXlS : Y

Gricl ploints *'ith thc sarlrc./ r'alue rlru' thc' sanrc ilrc

distancc' a\\'a\ trom side b (nrultiplc's ot' Aa ). For

arU U.j) position the (x'r1''),,.r,, is (Fig. l4):

L,u - a/f.

x' - b arcsin (tan(Aa(f-i))ltanA),

,'' - j La.

uvis - x.

4.1.3 Method aab

Side a is subdivided bV "f into La segments. From

side a perpendicular arcs are projected at each L,a

L'niliecl Approuch to Geodesic Domes

iute'n'al. From where these arcs intersect side c

allother arc is dropped to be perpendicular to side

b. All grid intersections lie on these projected per-

pendiculr arcs. For any (ii) position the (x" ,yo),..r., is

(Fig. 15):

L,a - a/f,

y" : jLa,

xo - arctan (cosy" sin(iAa)

(90o -a*ih'a)),

axis : y.

tan B/sin

Fig. 13.

0,6

=f o,s

aal

+ 0,4

0,3

0,2

o

vo,l

0,5

AA

0,4

o,2

o,0 1,0 2,O 3,0 4,O 5,0 6,0

constant constant constant constant constant constant

Method ab

f :6

Fig. 14.

b

0,6

-+

CI

(irl I

LlI

CI

ol

:+

I

(!l

-l

u1 I

=l

ol

,:*

cl

.'J I

i, I

=l

a zl

-2f I

(=l

el

'J1 I

CI

9+

EI

'2 I

OI

:+

el

-l

dl

=l

ol

,+ o,o x-"

b

22ri Intentational Journal of Space Structures Vol. 5 l/os. 3&4 1990

0,6

t o,s

aa I

+04

C.J. Kiuic'k

4.1.4 Method ba

Side b is subdivided bV "f into Lb arc segments.

From side c arcs are dropped to be perpendicuar to

side a such that the adjacent sides of these smaller

triangles are multiples of Lb. All grid intersections

lie on these arcs perpendicular to side a. Grid

points with the same f value are the same arc dis-

tance away from side a (multiples of Ab). For any

(ij) position the (xo,yo),,,, is (Fig. 16):

International Journal of Space Structures Vol. 5 Nos. 3&4 1990

b

Fig. 15.

Method ba

f :6

3ab

5ab

constant constant constant constant constant constailt

Fig. 16.

b

Lb - bt,,

xo : i&b,

yo : a arcsin (tan (L,u(f' - .il /tanB)).

axis : y.

4.1.5 Method bb

Side b is subdivided byl'into Ab arc se-qlrlents. At

each Lb distance a perpendicular arc is projectccl

until it intersects side c. Nl grid intersections lic orr

lt9

['nili*l .1ppy1111c,lt ttt Geodtsit, Donres

Fig. I7.

Fig. 18.

these proJected arcs. Grict points with the same Jr

value are the same arc clistance away from side b.

For any (ij) position the (xo,yo)o,,:, is lnig . l7):

xo : i&h,

y" _ arctan (sin(7A a)tanA),

exis - x.

4.1 .6 Merhod bha

Side b is subdividecl by ,f into Lb arc segmenm.

From side b perpenclicular arcs are projected at

each Lb interval. From where these arcs intersect

230

b

side c another arc is dropped to be pe{pendicular to

side a. Allgrid intersections rie on these proJected

pe{pendicular arcs. For any (ii) posiiion the

(x" ,yo)*i, is (Fig. l g):

Lb : b/f,

xo : i&b,

yo : arctan (cosro sn(ia,b) tanA/sin(90 b +

jLb)),

axis : y.

4.1.7 Method ca

side c is subdivided by _f into Ac arc segments.

Intemational Journal of space smtctures vol. Sl/os. 3&4 lgg0

C.J. Kitick

0,0 l,o 3,O 5,0 6,0

2,O

d { { { ::d ::d :J

constant constant constant constant constant constdlt

b

0,6

4,O

x" l'0

-+

f,l

ul

tf, I

CI

OI

:+

EI

el

al

5l

:+

CI

(trl

ul

(rt I

CI

a 8V

u7f

EI

-, I

6l

CI

ol

:+

e.t

(t, I

el

(nl

c. l

ol

i+

6l

*)l

al

cl

ol

"+

Method

f :6 cb

0,5

Fig. 20.

AA

o,4

o,3

o,2

3,0

b

0,6

0,5

o.4

0,3

o,2

o

vo,l

Fig. 21.

J,0

b

International Journal of Space Structures Vol. 5 Nos. 3A4 D90 231

From side c perpendicular arcs are dropped at

each A,c interval to be pe{pendicular with side a.

Atl grid intersections lie on these arcs perpen-

dicular to side a. The arc distances from any adja-

cent i values remains the same at allT values. For

any (i j) position the vo, !o),.o:, is (Fig. 19):

L,c - c/t

xo - arcsin (sin(iAc) sin B),

y' - a arctan (tan(A c(f-j)) cosB),

axis - y.

4.1.8 Method cb

Side c is subdivided by -f into Lc arc segments.

From side c perpendicular arcs are dropped at

each Ac interval to be perpendicular with side 6.

All grid intersections lie on these arcs perpen-

dicular to side D. The arc distances from any adja-

centT values remains the same at all i values. For

any (ij) position the (x", !")o** is (Fig.20):

Lc - c/f,

y" - arcsin (sin(7Ac) siM),

xo - b arctan (tan(Ac(f-i)) cos A),

axis : x.

4.1.9. Merhod cab

Side c is subdivided bV f into Ac arc segments.

From side c two arcs are dropped at each Ac inter-

val to intersect both side o and side b perpen-

dicularly. All grid intersections lie on these arcs

perpdndicular to both side a and b. For any (ij)

position the (x", !")r.rr, is (Fig.2l):

Lc - c/t

y" - a arctan (tan(A c(f-j)) cosB),

x' - b arctan (tan(Ac(f-i)) cos A),

xo : arctan (tanx' cosy"),

axis : y.

4.2 Method Ten - Radial Projection

Ivlethod ten is included due to its simplicity and

previous description.6 It was also used to provide a

reference frame for the l/s ratio comparisons. The

lcd triangle is divided by f in its planar form and

each intersection (ij) is projected radially until it

232

(lnifieel Approach to Geodesic Dontes

reaches the sphere surface. Thke the (x'J'z') posi-

tion on the plane and divide each of its com-

ponents by the distance to the center of the sphere

to find the projected (x,y,z) coordinates on the

sphere surface (Fig. 22).

Method Radial

22.

5. Comparative Analysis

Once we have a number of tessellations methods,

how do we ciifferentiate between them? There are a

number of commonly used yardsticks to measure

the appropriateness of a particular method to a

specific design problem. One of the most common

measures is that of complexity, which manifests

itself as uniqueness or lack of. Each (b,c) pair

generates a tessellation that has a maximum

potential for complexity constrained by the sym-

metry of the base geometry. Typically we would

like to minimize that complexity. For any (b,c) pair

there is a maximum number of different vertices,

edges, and faces that cAn be generated. The

approximation for the rnaximum number of dif-

ferent faces ffA is il3; the maximum number of dif:-

ferent edges (eO is t/2.

Alrother criteria for measuring a tessellation is

by comparing the ratio of longest to shortest edges,

or l/s ratio. For a regular icosahedron the lower

limit on the l/s rutio is l.l7 56 or the ratio of the long

side of an isosceles 54" , 72o , 54" triangle to the

short side. Tessellations with high l/s rutios tend to

be visually disjoint along the edges of the lcd

triangle, due to the pairing of long and short edges

in a single triangle resulting in rather non-

Intemational Journal of Space Structures Vol. 5 Nos. 3&4 1990

C.J. Kitick

equilateral conditions.

The ten methods presented were used to find

appropriate three dimensional coordinates for all

necess ary vertices for tessellations from t: I

(b :1,c -0) through t:256 (b : l6,c-0) inclusive'

Graphed against t are the number of faces fffi,,

edges @A,and the l/s ratio. It should be noted that

the certain values ofr appear more than once in the

sequence: example t:49 from (b,c) pairs (5,3)

and (7,0).'

5. I Different Faces

Graphs 1-10, derived from the generated data in

Thble 4, show the number of different face s ffA for

all values ofr for all ten methods. The maximum/i/

appears nearly as a straight line on the graphs. For

metho ds aa and bb, classes I and ll.Jd is a linear

function of b (on the order of {t). Table 2 provides

the formulas for determining the value of Jd tor

classes I and II and these two linear methods.

Table 2 - Different Faces (fd)

Class II

| zh-l

I

0b

90

80

70

Method Class I

oa I : b-

(sb-4)lt :b)

bb $b-l)/r :b)

Note: use integer Portion

112 1 33 1 48 1 69 t 89 201 223 243

t

Graph 1.

Method ab

r 12 133 148 169 189 201 223 243

t

Graph 2.

50

fdoo

30

20

10

0

60

50

fd oo

30

20

t0

0

Method aa

International Journal of space stntctures vol. 5 Nos. 3&4 1990 233

Method aab

Unified Approach to Geodesic Domes

97 112 133

t

Graph 3.

Method ba

79 97 112 133

t

Graph 4.

Method bb

97 1 1 2 133 148 1 69 189 201 223 243

t

Graph 5.

International Journal of Space Stntctures Vol. 5 Nos. 3&4 1990

231

C.J. Kitick

t

Graph 6.

Method ca

Graph 7.

Method cb

97 112 133

t

Graph 8.

Method bba

International Journal of Space Structures Vol. 5 Nos. 3&4 1990 235

Method

Unified Approach to Geodesic Domes

79 97 trr, 133

GraPh 9.

Method radial

97 112 131

t

Graph 10.

Method aa

112 133 l4E I ti9 I ou 4v I lLo

t

Graph 11.

International Journal of space structures vol- 5 Nos' 3&4 1990

236

C.J. Kitrick

r00

120

112 133 148 169 r89 201 223 243

t

Table 3 - Different Edges (ed)

5.2 Different Edges

Graphs ll-20, derived from the generated data in

Thble 5, show the number of different edges (ed) for

all values ofr for all ten methods. The maximumed

appears nearly as a straight line on the graphs. For

methods aa and bb, classes I and ll, ed is a linear

function of b (on the order of 1/t). Table 3 provides

the formulas for determining the value of ed for

class I and II and these two linear methods.

5.3 l/s Ratios

Graphs 21-30, derived from the generated data in

Thble 6, show the l/svalues over all values of/. In all

cases the value is becoming asymptotic, reaching a

maximum value. The graphs also show that there

is a wide fluctuation of l/s as all three classes are

mapped for a single method. Method cb, class II,

t40

120

Graph 12.

Method

aa

bb

Class I

I : b- I

(7b-8)tt : I <b <8

(7b - tt)tl : D ) 8

b

Class II

2b

2b

100

Note: use integer portion

which has already been shown to exhibit

asymptotically the minimum possible l/s ratio for

all triangular tessellations on the spheres'8 (the low

spikes on graph 28), also shows very small change

across all three classes giving the best overall l/s

ratios. The disadvantage here is that lh rutios do

not guarantee any minimizing of other factors

making its practical use rather limrted.

112 1 33 I 48 1 69 1 89 201 223 243

t

Method ab

Method aab

International Joumal of Space Structures Vol. 5 Nos. 3&4 1990

Graph 13.

8,5,129

9,4.1 33

11,1.133

1 0,3, 1 39

12,C.'144

7,7 ,147

11 ,2,147

8,5,148

9,5,151

10,4,156

12.r,157

1 1,3,163

8,7,169

13,0,1 69

9,6,1 71

12.2,172

10,5,175

11,4,181

13.1,183

12.3,1 89

8.8, r 92

9,7,193

10,5,1.96

14.0. /36

1 3,2,1 99

1 1 ,5.201

12,1,208

14 .1 ,211

9,8,217

13,3,217

10,7 ,219

11,a,223

15,0,225

14,2,228

12,5.229

13,4,237

15,1,241

9.9.24 3

10,8,244

11 .7,247

14,3,247

12.5.252

16,C,256

Table 4 : Oiflerent Faces (ld)

1

1

2

3

3

2

5

5

7

7

6

3

r0

11

7

13

13

15

4

17

9

17

19

21

17

10

23

25

5

21

27

11

25

31

3t

31

aa

13

35

6

37

37

25

30

14

38

43

43

45

45

47

15

7

49

44

51

45

53

55

57

17

41

51

32

61

6l

43

8

65

57

18

67

67

43

71

lo

73

73

75

19

65

77

79

81

I

70

83

83

39

21

cab radial

11

238 Intemational Joumal of Space Structures Vol. 5l{os. 3&4 1990

b,c,t

1,0,1

1,1,3

2,0.4

2,1,7

3,0,9

2,2,12

3,1 ,13

4,0,16

3,2,19

4.1 ,21

5,0,25

3,3,27

4,2,28

5,1,31

6,0,36

1,3,37

5,2,39

6,1 ,43

4,4,48

5,3,49

7 ,A,49

6,2,52

7,1 ,57

5,4,61

6,3,63

8.0,64

7,2,67

8,'1,73

5,5,75

6.4,76

7.3,79

9.0,81

9,2,84

6,5.91

9.1,91

7,4,93

8,3,97

1 0,0,1 00

9,2.1 03

6,6.108

7,5.109

10.1 .1 1 1

8,4,112

9,3.1 17

1 1,0,121

10,2,124

7,6,127

8,5,1 29

9,4,133

11.1.133

10,3,139

12,0,144

7 ,7 ,147

11 ,2,1 47

9,6,148

9,5.1 5 1

1 0,4,1 55

12,1,157

1 1,3,163

8,7,1 69

1 3,0.1 69

9,6,171

12,2,172

10,5,175

11,4,181

13,1 ,1 83

1 2,3,1 89

8,8,192

9,7,193

10,5,195

14,0,196

13,2.199

11,5,201

12,4,208

14,1 ,211

9,8,217

13,3,217

10.7 .219

11,6,223

15,0,225

14,2,228

12,5.229

13,4,237

1 5.1 ,241

9,9,243

10 ,8,244

1't ,7,247

14,3,247

12,6,252

16,0.255

2

1

2

2

3

1

2

2

4

3

4

7

4

10

11

5

b

13

16

6

I

0

2

8

5

7

3

9

1

3

I

4

7

0

3

0

9

5

46

46

47

49

10

52

12

55

56

32

41

11

53

54

65

67

67

70

12

14

74

62

76

64

79

82

85

13

57

72

41

91

92

59

16

97

81

l4

100

101

57

06

08

09

10

't2

15

93

15

19

21

18

00

24

124

51

l6

UniJiecl Approach to Geodesic Domes

Table 5 : Oillerent Edges (ed)

aab bb ba bba ca cb radial

122

123

123

125

60

C.J. Kitick

97 112 133

t

Graph 14.

97 112 I 33

t

Graph 15.

1 13 28 48 63 79 97 112 133

t

Graph 16.

International Journal of space structures vol. 5 Nos. 3&4 1990

Method ba

Method bb

Method bba

239

I\{e thod cil

Liniliul .l\tprouch to Geoclesic Domes

97 112 13

t

Graph 17.

Method cb

97 112

Graph

Method

18.

cab

97 112 I 33 1 48 1 69 1 89 201 223 243

t

Graph 19.

Intentational Joumal of Space Structures Vol. 5 Nos. 3&4 1990

240

Method radial

C.J. Kitick

r40

120

100

80

60

ed

1.54

1.5

1.46

1.42

r.38

l/s1.34

1.3

1.26

1.22

1.18

1 .14

1.54

1.5

r.46

1.42

r.38

l/s

' 1.34

1.3

1.26

1.22

l.'18

1.14

t

Graph 20.

1 08 124 114

t

Graph 21.

52 7g 91 108 124 1 4/t

t

Graph 22.

Vol. 5 Nos. 3&4 1990

40

20

0148 169 189 201

r57 175 196 217 229 217

r75

r57 217

37

Method aa

Method ab

International Journal of space structures

196

241

Method aab

1.54

1.5

1.46

1.42

r.38

I / Sr.sr

1.3

1.26

1.?2

r.l8

l.t 4

1.54

1.5

1.46

1.42

r.38

I / Sr.gr

r.3

1.26

1.22

1.18

l.t 4

1.54

1.5

1.45

1.42

1.38

I / t,.rn

t.3

1.26

1.22

1.18

1 .14

1 08 121 1 44 157 175 1 96 217

t

Graph 23.

Method ba

Uni,fiecl Approach to Geodesic Donres

la4 157 175 196 217

Graph 24.

to8 121 144 157 175 196 217 229 217

t

Graph 25.

Intentational Jountal of space structures Vol.

Method bb

242 5 Nos. 3&4 1990

Method bba

C.J. Kitick

l/s

1.54

1.5

1.46

1.42

1.38

1.34

1.3

1.26

1.22

1 .18

t.l4

1.54

1.5

1.46

1.12

1.38

l/s

' 1.34

1.3

1.26

1.22

1.18

1 .14

108 124 144

t

Graph 26.

1 0E 124 144

t

Gruph 27.

157 .175 196 217 229 247

157 175 196 217 229 217

Method cb

1.54

1.5

1.46

1.12

1.38

l/s1.3,1

1.3

1.26

1.22

1.18

l.l4 t08 121 144 157

t

International Journal of Space Structures

Graph 28.

Vol. 5 Nos. 3&4 1990

Method ca

175 196 217 229 247

243

Method cab

Unified Approach to Geodesic Domes

r.5

1.46

1.12

1.38

I/sr.3a

r.3

1.26

1.22

1.r 8

1.14

r.54

r.5

1.t36

1.12

r.38

l/sr.34

t.3

r.26

1.22

6. Cross Reference

Over many years there have been numerous

individuals who have pursued the problem of

spherical subdivision. Much of the work has gone

unpublished and what has been published has

typically been rather isolated for long periods of

time. The nine spherical methods presented in this

paper are not a complete set of those available but

they do not include those most used in praciice

(Methods aa and bb). The set of nine are actually

the author's extension of the comm on aa and bb

methods when viewed in their spherical form and

in context of the underlying grid structure. The

following table is the author's attempt to cross-

reference previous published work in relation to

the ten presented.

24

91 r08 124 t{1 t57 175 196 217 229 247

t

Grupln29.

Method radial

I 08 121 I {'l 157 I 75 I 96 217 229 217

t

Graph 30.

aa

ab

aab

ba

bb

Table 7 - Cross Reference

Method Reference [Notation: Author (description) (classes)]

Scheele (Class II)

Clinton6 lMethod 6) (Class II)

:

Stuatlo (Class II)

Tarnail I (Class I)

Pavlovl'(o Method) (Class I&ID

Clinton6 6'tethod 6) (Class I)

bba

ca

cb Kitrick3 (Class II)

cab

radial Clintonl lMethod I&IV) (Class I&ID

Intemational Journal of Space Stntctures Vol. 5 Nos. 3&4 1990

C.J. Kitick

Table6:Usdata

b.c.t ail

1,0,1 1.00000

1,1,3 1.11359

2,0,4 1.13083

2,1 ,7 1 .17356

3,0.9 1 .20164

2,2,12 1.21851

3,1,13 1.21184

4,0,16 1.25956

3,2,19 1.25570

4.1 ,21 1.28220

5.0,25 1.30184

3,3,27 1.26792

4,2,28 1.29284

5,1 .31 1.3'14O4

6,0,38 1.32593

4,3,37 1.29368

5.2,39 1.31849

6.1 ,43 1.33630

4.4,49 1.29476

5,3,49 1.31876

7 ,O,49 1 .34629

6,2,52 1.33863

7,1,57 1.35408

5.4,61 1.31448

6,3,63 1.33726

8,0,64 1.38335

7,2,67 1.35507

8.1 ,73 1.36786

5,5,75 1.31147

6,4,76 1 .33339

7 ,9,79 1.35270

9,0,81 1.37433

8,2,84 1.36764

6.5,91 1.32747

9.1 .91 1 .37888

7,4,93 1.34820

9,3,97 1.36532

10,0,100 1.38436

9,2.103 1.37867

6,6,108 132293

7,5 ,1 09 1 .34279

10,1,111 1.38793

8,4,112 1.36069

9,3,117 1:37588

1 1 ,0,1 21 1 .39342

10,2,124 1 .38746

7,6,127 1.33632

8.5,1 29 1 .3551 I

9,4.133 1.37135

1 1 .1 .133 1.39561

10.3,139 1.38468

12,0,144 1.39963

7,? ,147 1.33105

11 ,2,147 1.39500

8,6,148 1.34921

9,5,151 1.36591

10.4.156 1.38034

12.1,157 1.40204

1 1 ,3,1 63 1.39237

8,7,169 1,34271

13,0.1 69 1 .40556

9,6.171 1 .35994

12,2,172 1 .40141

1 0,5,1 75 I .3751 0

1 1 ,4,181 1.38820

13,1 ,1 83 1.40762

12,3,'l 89 1 .39872

8,8.192 1.33727

9,7,193 1.35388

10,6,1 96 1 .36933

14,0,196 1.41118

13.2,199 1.40689

1 1,5,201 1.38320

12,4,208 1.39502

14,1,211 1 .41242

9,8,217 1.34755

13,3,217 1.40456

.10,7 ,219 1.36336

1 1 ,6,223 1.37760

15,0,225 1.41513

14,2,228 1.41175

12,5,229 1.39023

13,4,237 1.40091

15.1 .241 1 .41665

9,9,243 1.34214

10,8,244 1.35740

11 ,7,247 1.37179

14,3,247 1.40948

12,6,252 1.38485

16.0.256 1 .41905

ab aab

1.00000 1.00000

1.r1359 1.11359

1.13083 1.13083

1.18183 1.19856

1.19157 1.24946

1.16113 1.21851

1.21181 1.26452

1.22320 1.28427

1.19353 1.30365

1.23680 1.32964

1.24762 1.37c30

1.18923 1.277A7

1.21108 1.3206t

1 .24315 I .36684

1.24S5t1 1.38673

1.20977 1.34228

1.24383 1.38847

1.25917 1.40010

1.17202 1.31656

1.20654 1.38217

1.27726 1.43415

1.23899 1.39743

1.25913 1.12528

1.21802 1.37364

1.24339 1.40326

1.27429 1.43904

1.27157 1.43692

1.27672 1.44804

1.17818 1.34730

1.21106 1.38227

1.23958 1.41517

1.29328 1.46690

1 .26829 1 .44191

1.22025 1.38651

1.27877 1.45948

1.24941 1.42016

1.26717 1.44217

1.29105 1.46964

1.28830 1.46704

1.18737 1.36331

1 .21546 1.39640

1.29037 1.47419

1.24493 1.42596

1.26513 1.44955

1.30332 1.48793

1.28593 1.47007

1.22538 1.39977

1.247?5 1.42502

1.27029 1.45194

1.29137 1.48172

1 .28270 1.46827

1 .30162 1.48965

1.19412 1.37809

1.29926 1.48725

1.21951 1.40561

1.24396 1.43150

1.25717 1.45587

1.29958 1.4919E

1.28211 1.47309

1.22819 1.40607

1.31020 1.50255

1 .24901 1.43238

1.29748 1.48923

1.26S12 1.45274

1.28472 1.47439

1.30011 1.49729

1.29345 1.48668

1.19927 1.38685

1.22130 1.41251

1.24488 1.43679

1.30888 1.50371

1.30691 1.50165

1.26513 1.45703

1.28241 1,477'.13

1.30622 1.50483

1,22885 1.41332

1.29355 1.48999

1.24727 1.43458

1.26638 1.45679

1.31521 1.51331

1.305s3 1.50302

1.27973 1.47317

1.29506 1.49083

1 .30650 1.50877

1.20335 1.39529

1.22379 1.41775

1.24444 1.43906

1 .30126 1 .50025

1.26333 1.4601 1

1.31416 1.51414

bb ba bba

1.00000 1.00000 1.00000

1.11359 1.11359 1.11359

1.13083 1.13083 1.13083

1.14872 1.15205 1.17867

1.15561 1.15561 1.15581

1.24287 1.19636 1.21?87

1.207 12 1.23A77 1.25548

1.21330 1.16433 1.21330

1.2127 4 1.22739 1.28484

1.21eSo 1.25075 1.28257

1.200n 1.16837 1.21353

1.28284 1.30666 1.3734+

1.23411 1.26208 1.31974

1.25212 1.28739 1.36375

1.19183 1.19204 1.22989

1.25792 1.29925 1.37253

7.2*172 1.27431 1.33141

1.L*52 1.2es63 1.35218

1.30096 1.33039 1.,10695

1.25831 1.33205 .1.40630

1.242'.t9 1.21105 1.29437

1.27991 1.29669 1.97987

1.24699 1.31 108 1.36513

1.27920 1.32339 1.40508

1.27417 1.33333 1.40354

1.23332 1.22658 1.29400

1.27143 1.30729 1.37631

1.27664 1.31590 1.40210

1.31 132 1 .36706 1 .45028

1.2729A 1.34821 1.42944

1.29170 1.35468 1.44140

1.22682 1.23947 1.30031

1.26519 1.30s87 1.37509

1.29192 1.35545 1.44188

1.26661 1 .32541 1.39636

1.29384 1.35238 1.42968

1.28288 1.35216 1.42945

1.25867 1.25033 1.33946

1.28930 1.32079 1.41120

1.31793 1.37792 1.46475

1.28664 1.37837 1.46421

1.25826 1.32805 1.39292

1.29887 1.36102 1.45087

1.27809 1.36277 1.43609

1.24949 1.25950 1.33881

1.28035 1.323s2 1.40107

1.30050 1.36692 1.45705

1.30512 1.37345 1.45688

1.29105 1.36556 1.44846

1.28106 1.33425 1.42476

1.29817 1.36230 1.45356

1.24220 1.28760 1.33685

1.32251 1.39621 1.4861 1

1.27300 1.32816 1.40085

1.29603 1.38653 1.47529

1.30618 1.38831 1.48002

1.28907 1.36418 1.44505

1.27193 1.33624 1.41403

1.28981 1.36902 1.45091

1.30672 1.38485 1.477AO

1.26617 1.27457 1.36856

1.312s0 1.38588 1.47212

1.29141 1.33195 1.42509

1.296s6 1.38229 1.46973

1.30471 1.37198 1.46493

1.26430 1.34051 1.41214

1.28321 1 .36734 1.44365

1.32585 1.40240 1 .49410

1.30321 1.40260 1.49372

1.31093 1.39248 1.48592

1.25797 1.28069 1.38386

1.28335 1.33364 1.41713

1.29998 1.39170 1.17602

1.29674 1.37056 1.45555

1.28270't.34191 1.43426

1.31145 1.39188 1.48572

1.29929 1.37225 1.46587

1.31818 1.39593 1.48578

1.30219 1.39350 1.48309

1.25112 1.28e12 1.36227

1.27549 1.33706 1.41421

1.30973 1.38846 1.48288

1.29122 1.37376 1.45544

1.27483 1.34516 1.42718

1.32841 1.41335 1.50679

1.30849 1.40751 1.50021

1.31470 1.40798 1.SO2S2

1.29175 1.37108 1.45600

1.30726 1.39368 1.48136

1.27033 1 .29096 1.38580

cb cab radial

1.00000 1.00000 1.00000

1.11359 1.11359 1.11359

1.13083 1.13083 1.13083

1.15002 1.14972 1.22'.116

1.15663 1.15663 1.18300

1.15997 1,15997 1.26645

1.20104 1.21874 1.29539

1.17850 1.17850 1.28333

1.17827 1.21569 1.35273

1.20996 1.26795 1.31695

r.19603 1.22075 1.32022

1.1 6863 1 .25517 1 .37358

1.20125 1.?5732 1.38416

1.22220 1.31 034 1.36635

1.20582 1.24035 1.33255

1.18610 1.26659 1.41652

1 .21 101 1.27248 1.39182

1.22188 1 .3170S 1r3t1520

1.17166 1.27268 1.42782

1.20248 1.31079 1.43297

1.21 186 1 .26372 1.36421

1 .21636 1.30180 1.41838

1.22861 1.33952 1.39008

1.18633 1.28249 1.45316

1.21009 1.31773 1.43591

1.21587 1.27568 1.37828

1.22078 1.31252 1.42829

1.22628 1.34138 1.40906

1.17307 1.31451 1.46012

1.19832 1.31309 1.46309

1.21832 1.34390 1.45185

1 .21868 1 .29045 1 .3831 6

1.22326 1.32535 1.42ss1

1.18551 1 .31 144 1.47674

1.23105 1.35539 1.11704

1.20631 1.32102 1.46422

1.22123 1.34703 1.45743

1.22073 1.29847 1.39818

1.22665 1.33327 1.44127

1.17383 1.33165 1.48144

1.19626 1.33425 1.48334

1.22833 1.35539 1.41880

1.21239 1.33894 1.47464

1.22620 1.36345 1.45706

1.22228 1.30863 1.40548

't,22812 1.33974'.|.44570

1.18457 1.32482 1.49316

1.20326 1.33694 1.48371

1.21663 1.34708 1.47800

1.23212 1.36516 1.42863

1.22691 1 .36443 1.46468

1.22348 1.31437 1.40803

1.17429 1.35269 1.49653

1.22968 1.34568 1.44570

1.19364 1.33521 1.49784

1.21031 1.35709 1.49098

1.22036 1.35484 1.47696

1.22941 1.36440 1.43296

1.23037 1.37546 1.46709

1.18372 1.34121 1.50523

1.22443 1.32178 1.41674

1.20047 1.33989 1.49787

1.23062 1.34946 1.45217

1.21431 1.35846 1.49310

1.22333 1.36221 1.48212

1.23261 1.37173 1.43377

'a.23002 1.37532 1.46617

1.17459 1.36163 1.5A776

1.19202 1.34626 1.50872

1.20632 1.35425 1.50320

1.2252A 1.32609 1.42118

1.23134 1.35397 1.45460

1.21903 1.37497 1.49183

1.22523 1.36516 1.48337

't.23005 1.37063 1.43974

1.18299 1.34866 1.51447

1.23268 1.38320 1.47050

1.19816 1.34729 1.50859

1.21073 1.36433 1.50459

1.225W 1.33173 1.42273

1.23199 1.35651 1.45433

1.22122 1.37888 1.49546

1.22719 1.37286 1.48201

1.23283 1.37641'.t.44245

1.17480 1,37434 1.51644

1.19032 1.34785 1.51716

1.20425 1.36867 1.51254

1.23180 1.38247 1.47173

1 .21468 1.37646 1.50328

1.22636 1.33509 1.42841

1.00000

1.1 1359

1 .1 3083

1.1 493 1

1.15598

1.15997

1.2a6tt

1.16480

1,2A734

1.24t58

1.1 8731

1.25517

1.2450E

1.28344

1.21179

1,26211

1.25904

1.29402

1.28998

1.30137

1.231 18

1.28488

1.31 192

1.27882

1.30710

1.24683

1.29691

1.31788

1.30052

1.30745

1.32981

1.25970

1.30699

1.30204

1.32853

1 .31467

1.33335

1.27046

1.31 684

1.30712

1.32934

1.33230

1.32964

1.347'.17

1.27958

1.32106

1.30950

1.33087

1.33302

1.33932

1.34927

1.28740

1.32207

1.32908

1.33203

1.34894

1.34351

1.34190

1.35817

1.32208

1.29418

1.33612

1.33235

'!.34995

1 .35160

1.34685

1.35943

1.32577

1.34330

1.34850

1.3001 1

1.33743

1.36221

1.35276

1.34873

1.32625

1.36546

1.34344

1.35421

1.30535

.34043

.36275

.36003

.35239

.33462

.34475

.35868

.36623

.35965

.31000

International Journal of Space Structures Vol' 5 Nos' 3&4 1990 245

7. Conclusions

A practical approach has been presented for

understanding all three classes of geodesic domes

in one general framework. This framework gives

us the abiliry to extend class specific tessellation

methods to cover all classes. Also presented were

ten methods to generate tessellations, including

the ones most often used by designers. A small

comparative study was provided to illustrate the

effect of the different methods on the different

number of faces, edges and l/s ratios generated

for each.

References

l. COXETER, H.S.M., "Virus Macromolecules and

Geodesic Domes", A Spectntm of Mathematia, €d. J.C.

Butcher, Auckland University Press and Oxford

Universiry Press, 1972, PP. 98-107.

2. WENNINGE& M. J., Spheical Models, Cambridge

University Press, 1979, p. 98-100, 120'l2l-

3. KITRICK CJ., "Geodesic Domes",,Srructural Tbpologt,

University of Montreal, 11, 1985, pp. I 5-20.

Unt-liecl Approach to Geodesic Domes

TARNAI, T., "Geodesic Domes with Skew Networks",

Spheical Grid Structures, ed. T. Tarnai, Hungarian

Institute for Building Science. 1987. .

IVIAIQ{I, 8., "On Polyhedra with Approximately Equal

Edges", Spherical Grid Stntctures, ed. T. Thrnai,

Hungarian Institute for Building Science, 1987.

CLINTON, J.D., Advanced Stnrctural Geometry Studies,

Part I, Polyhedral Subdivision Concepts for Structural

Applications, NASA CR-1734, Sept. 1971.

GOLDBERG, M., '.[ Class of Muiti-Symmetric

Polyhedra", Tbhoku Mathematical Journal, 43, Tokyo,

Japan, 1937, pp. lM-108.

MAKAI, E., and TARNAI, T., "IJniformity of Networks

of Geodesic Domes", Spheical Gid Stntctures, ed. T.

Tarnai, 1987, Hungarian Instinrte for Building Science.

SCHEEL, H., "Geodesic Surface Division", The Cana'

dian Architect, 14, No. 5, 1969, pp. 61-66.

STUART, D.R., A Report on the Tiiacon Gridding Sys-

tem for Spherical Surface, Skybreak Carolina Corpora-

tion, 1952.

TA-RNAI, T., "spherical Grids of Tiiangular Nenvofk",

Acta Tbchnica Acad. Scl Hungar., 7 6, 197 4, pp- 307 -

336.

PAVLOV, G.N., "Compositional Form-shaping of

Crystal Domes and Shells", Spherical Gid Stntctures, ed.

T. Tarnai, 1987, Hungarian Institute for Building

Science.

4.

5.

6.

7.

8.

9.

10.

11.

12.

246 International Journal of Space Stntctures Vol. 5 Nos- 3&4 1990