The
following figures analyze the stability of the previous polygons using
Euler's equation. 




a) hexagon 
b) pentagon 
c) square 
d) triangle 







6<2(6) 3 
9=2(6) 3 
5<2(5) 3 
7=2(5) 3 
4<2(4) 3 
5=2(4) 3 
3=2(3) 3 
unstable 
stable 
unstable 
stable 
unstable 
stable 
stable 
(need + 3 

(need + 2 

(need + 1 


members) 

members) 

members) 



Fig. 123  Analyzing the stability of polygons
with Euler's equation M = 2 J  3 


Notice
that the equation only refers to how many structural elements (members and
joints) 
each
polygon has. That is, it describes the topology of the polygonal
structure regardless 
of the
length of its members or the shape of the polygon (regular or irregular,
segmented 
or not).
This is the same type of topological analysis that was used in the
previous lesson 
to
describe the structure of crystals. 

EXERCISE: Derive Euler's equation for stable polygons using simple 

algebra or a computer graphing program. 


Euler's
equation gives a fairly reliable indication of the minimum number
of members 
required
for stable polygons. Obviously stable polygons have more
members than the 
required minimum. For example the square can be stabilized with
seven or eight 
members
instead of the required minimum of five. And eleven or twelve
members can be 




7 > 2 (4)  3 
8 > 2 (4)  3 
11 > 2 (6)  3 
12 > 2 (6)  3 
redundant 
redundant 
redundant 
redundant 




Fig. 124  Stable but redundant polygons
(demonstration models) 


used to stabilize the
hexagon instead of a minimum of nine. Such structures are said to be 
stable but redundant. 
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Page 80 
Building stability  Twodimensional stability 

