5.1a
Taking square ABCD project the inner surface divisions by circular arcs
onto the square's linear base. With C as centre and radius CA, project base
line EG. Project line CD in a similar manner, giving line DF. Join AE and AG to
find three similar triangles.
When AB=1, CA=Ö2, ED=Ö2-1, DG=Ö2+1
Have a
go!
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5.1b
Taking square ABCD, rotate the semi-diagonal AX to mark E and F on the
extended base line.
When AB=1,
XA=(Ö5)/2,
ED=(Ö5)/2
- 1/2,
DF=(Ö5)/2 + 1/2
Have a
go!
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5.1c
With D as centre, swing arc EG. Project GJ parallel to DC, defining the
rectangle DCHG and square CFJH.
When AB=1,
DFJG=ABCD=1
ABHG is a Golden Rectangle.
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5.2
Draw a double square and extend the dividing line EF. Wirh G as centre and
a semidiagonal GA are radius, swing an arc intersecting EF at H.
When
AB=1,
GE=1/2, GH=GA=Ö5/2
FH=(1+Ö5)/2.
Thus the Golden Rectangle JBFH arises out
of the double square through its Ö5
rectangle.
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5.3
From square ABFE construct HK=Ö5. With E
and F as centres and radius FN, swing arcs HN and KN. With E andF as centres
and radius FB, swing arcs to intersect arcs HN and KN at O and P respectively.
Connect F, E, O, N, P to foerm a pentagon.
The side of a pentagon is
in relation to its diagonal as 1:(1+Ö5)/2.
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