construction

MATHS BITE: Heptadecagon

A heptadecagon (or a 17-gon) is a seventeen sided polygon.

File:Regular polygon 17 annotated.svg

Regular Heptadecagon | Wikipedia

Constructing the Heptadecagon

In 1796, Gauss proved, at the age of 19 (let that sink in…) that the heptadecagon is constructible with a compass and a straightedge, such as a ruler. His proof of the constructibility of an n-gon relies on two things:

  • the fact that “constructibility is equivalent to expressibility of the trigonometric functions of the common angle in terms of arithmetic operations and square root extractions“;
  • the odd prime factors of n are distinct Fermat primes.

Constructing the regular heptadecagon involves finding the expression for the cosine of  2\pi /17 in terms of square roots, which Gauss gave in his book Disquistiones Arithmeticae:

{\displaystyle {\begin{aligned}16\,\cos {\frac {2\pi }{17}}=&-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+\\&2{\sqrt {17+3{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {34+2{\sqrt {17}}}}}}\\=&-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+\\&2{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}.\end{aligned}}}

Source: Wikipdia

An explicit construction was given by Herbert Willian Richmond in 1893.

Regular Heptadecagon Using Carlyle Circle.gif

Source: Wikipedia

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The Sierpinski Triangle

The Sierpinski Triangle, or Sierpinski Sieve, is a fractal described by Polish Mathematician Sierpinski in 1915, although it appeared in Italian art from the 13th century. It has an overall shape of an equilateral triangle, and is subdivided recursively into smaller equilateral triangles.

Sierpinski sieve from rule 90

Source: Wolfram Mathworld

Constructing a Sierpinski Triangle

STEP 1:

Start with an equilateral triangle.

STEP 2:

Connect the midpoints of each side, hence dividing it into 4 smaller congruent equilateral triangles.

Triangle

STEP 3:

Now cut out the triangle in the centre.

Step One

STEP 4:

Repeat steps 2 and 3 with each of the remaining smaller triangles.

[Sierpinski Triangle]

Properties

 

If we let N_n be the number of black triangles after iteration nL_n be the length of a side of a triangle, and A_n be the fractional area which is black after the nth iteration, then:

Screen Shot 2016-11-27 at 6.53.59 PM.png

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