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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
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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
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4.
5.
6.
7.
8.
9.
10.
11.
12.
246 International Journal of Space Stntctures Vol. 5 Nos- 3&4 1990
