Prerequisites : Ruler and ompass rules and methods
Objectives : • study the onstru tibility of regular polygons. Materials : • Ruler • Compas • Task sheet • Hints for the • Beamer
pentagon, 15-gon and 16-gon.
1 – The easy ones
20 mins
Students work in pairs. They have to nd the minimal number of a tions needed to draw an equilateral triangle, a square, a regular hexagona, and a regular o togon. This part is marked over 10.
2 – The regular pentagon
20 mins
The se ond task is to onstru t a regular pentagon. This part is marked over 10. Progressive hints are available on demand. Every hint asked by a group takes one point of the nal mark.
3 – The regular pentadecagon
15 mins
The third task is to onstru t a regular 15-gon. One hint may be given : use a pentagon and an equilateral triangle. This par is not marked
4 – The regular heptadecagon
20 mins
The third task is to onstru t a regular 17-gon. A onstru tion proto ol is given to ea h student and has to be arried out. The end result is also marked over 10.
5 – Constructible polygons
20 mins
Ea h group has to list all onstru tible polygons with a number of sides less than or equal to 20.
6 – Lecture : Some results about constructible polygons
15 mins
A qui k history of the problem of onstru tible polygons, in luding Eu lid's methods, Gauss' results and Gardner's link with the Sierpinski's binary sieve.
Constructible polygons
Season Episode Document
3 07 Answer sheet
Part 1 – Construct some easy regular polygons Equilateral triangle
Square
Regular Octogon
Regular hexagon
Season 3 • Episode 07 • Constructible polygons
2
Construction of the pentagon Try to nd a way to onstru t a regular pentagon. If you don't manage, go to the tea her's desk and ask for a hint. The hints are progressive and ost 1 point over 10 ea h.
Construction of the pentadecagon Try to nd a way to onstru t a regular pentade agon (with 15 equal sides). If you don't manage, go to the tea her's desk and ask for a hint. There is only one available hint and this part is not marked.
Season 3 • Episode 07 • Constructible polygons
3
The heptadecagon : a construction protocol Follow the following instru tions to onstru t a regular heptade agon. 1. Given an arbitrary point O , draw a ir le entered on O and a horizontal diameter drawn through O . 2. Call the right end of the diameter dividing the ir le into a semi ir le P1 . 3. Constru t the diameter perpendi ular to the original diameter by nding the perpendi ular bise tor OB , with B at the top of the ir le. 4. Constru t J a quarter of the way up OB . 5. Join JP1 and nd E on line segment OP1 so that ∠OJE is a quarter of ∠OJP1. 6. Find F on line OP1, but on the other side of O , so that ∠EJF is 45 degrees. 7. Constru t the semi ir le with diameter F P1 , on the same side as J . This semi ir le
uts OB at K . 8. Draw a semi ir le with enter E and radius EK , on the same side as B and with both endpoints on OP1 . This uts the line segment OP1 at N4 . 9. Constru t a line perpendi ular to OP1 through N4 . This line meets the original semi ir le at P4 . 10. You now have points P1 and P4 of a heptade agon. Use P1 and P4 to get the
remaining 15 points of the heptade agon around the original ir le by onstru ting P1 , P4 , P7 , P10 , P13 , P16 , P2 and so on. 11. Conne t the adja ent points Pi for i = 1 to 17, forming the heptade agon.
4
Season 3 • Episode 07 • Constructible polygons
The constructible polygons 1. List all regular polygons with 20 or less sides that you think are onstru tible withe
ruler and ompass. Explain in a few words ea h onstru tion.
2. A Fermat prime is a prime number of the form Fn = 22 +1, where n is a nonnegative n
integer. Compute the rst ve Fermat primes.
3. There is a onne tion between the onstru tible polygons and the Fermat prime.
Try to nd it.
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Season 3 • Episode 07 • Constructible polygons
Document 1 Hints for the onstru tion of the regular pentagon
1.
Draw a ir le in whi h to ins ribe the pentagon and mark the enter point
2.
Choose a point
O.
A
on the ir le that will serve as one vertex
of the pentagon. Draw a line through
3.
O.
the point
A.
Mark its interse tion with one side of the ir le as
B.
4.
Constru t the point
5.
Draw a ir le entered at
C
as the midpoint of
interse tion with the line the point
O
and
B.
C through the point A. Mark its OB (inside the original ir le) as
D.
Draw a ir le entered at
A
Draw a ir le entered at
E
D . Mark its
ir le as the points E and F .
through the point
interse tions with the original
7.
and
Constru t a line perpendi ular to the line OA passing through point
6.
O
through the point
A.
Mark its
other interse tion with the original ir le as the point
8.
Draw a ir le entered at
F
through the point
A.
G.
Mark its
other interse tion with the original ir le as the point
number of distinct Fermat primes. Gauss conjectured that this condition was also necessary. It was proved by Pierre Wantzel in 1837. Episode 07 â Constructible ...
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