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Mechanism for Constructing Regular Polygons

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n-gon
7
move
show polygon
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This Demonstration shows a hinged mechanism for constructing regular
n
-gons,
n=4,…,10
. The mechanism is constructed as follows. The fixed hinges are at
A
,
B
,
C
,
G
,
H
,
I
, and
J
, while the hinges at
D
and
E
slide along
AG
and
BI
respectively, and
O
is the midpoint of
AB
. The mechanism contains congruent parallelograms
ABHG
and
BCJI
and two isosceles trapezoids
DABC
and
DJIE
. This ensures that the sides
AB
,
BC
,
CD
, and
DE
have the same length and that angles
ABC
,
BCD
, and
CDE
are equal.
A square is constructed by moving the mechanism to make
E
coincide with
A
. To draw the red perpendicular to
AB
at
O
, which is needed in the construction of the pentagon, hexagon, heptagon, and octagon, keep the mechanism rigid and slide it along
AB
until a vertical line is at
O
(A,B)
. The nonagon and decagon need the red line to be inclined at 60° and 36°; these angles come from the hexagon and the pentagon.
To construct a heptagon (or 7-gon), keep
AB
in place and turn the mechanism until
E
is on the vertical red line. Four sides of the heptagon are determined. Freeze the hinges and rotate the rigid mechanism so that the rod
AB
coincides with the previous location of
DE
. That determines the remaining three sides. That the pentagon
OBCDE
forms one half of the regular heptagon follows from the fact that the polygonal line
ABCDE
is symmetric with respect to
CK
.
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