Outline Examples of geometrical star designs Angle sum of a - - PowerPoint PPT Presentation

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Outline Examples of geometrical star designs Angle sum of a - - PowerPoint PPT Presentation

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Outline Examples of geometrical star designs Angle sum of a triangle and other polygons Properties of a pentagram A method for constructing other stars Connections with number theory and complex numbers Angle sum of

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Outline

  • Examples of geometrical star designs
  • Angle sum of a triangle and other polygons
  • Properties of a pentagram
  • A method for constructing other stars
  • Connections with number theory and complex

numbers

  • Angle sum of arbitrary stars
  • Turtle programming with Python
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Examples – 5 points

Somalia

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Examples – 5 points

Senegal

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Examples – 5 points

Timor-Leste

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Examples – 5 points

Saint Kitts and Nevis

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Examples – 5 points

Morocco

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Examples – 5 points

Ethiopia

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Examples – 5 points

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Examples – 6 points

Israel

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Examples – 6 points

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Examples – 6 points

2 disconnected triangles

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Examples – 7 points

Australia

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Examples – 7 points

Sheriff badge, Suffolk County, NY

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Examples – 7 points

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Examples – 7 points

2 different types

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Examples – 8 points

Azerbaijan

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Examples – 8 points

14th century Iranian tile

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Examples – 8 points

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Examples – 8 points

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Examples – 8 points

connected disconnected

2 different types

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Examples – 9 points

Baha’i symbol Slipknot logo Baha’i symbol

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Examples – 9 points

connected disconnected connected

3 different types

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Examples – other

Aruba: 4 points Nauru: 12 points Malaysia: 14 points

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Examples – other

Aruba: 4 points Nauru: 12 points Malaysia: 2 x 7 points Marshall Islands: 24 pts

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D E A B C 𝛽 𝛾 𝛿

Angle sum of a triangle – Method 1

  • Consider triangle 𝐵𝐶𝐷 with

angles 𝛽, 𝛾 and 𝛿

  • Construct line-segment 𝐸𝐹

through 𝐵 parallel to 𝐶𝐷

  • By alternate angles

𝛾 = ∠𝐵𝐶𝐷 = ∠𝐸𝐵𝐶 𝛿 = ∠𝐵𝐷𝐶 = ∠𝐹𝐵𝐷

  • By adjacent angles in a

straight angle 𝛽 + 𝛾 + 𝛿 = 180°

𝛿 𝛾

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180° − 𝛾 180° − 𝛿 180° − 𝛽 A B C 𝛽 𝛾 𝛿

Angle sum of a triangle – Method 2

  • Consider triangle 𝐵𝐶𝐷 with

angles 𝛽, 𝛾 and 𝛿

  • Extend sides to form exterior

angles 180° − 𝛽 etc.

  • Shrink 𝐵𝐶𝐷 to a single point

with three exterior angles

  • By adjacent angles in a

revolution 180° − 𝛽 + 180° − 𝛾 + (180° − 𝛿) = 360°

  • Hence

𝛽 + 𝛾 + 𝛿 = 180°

180° − 𝛾 180° − 𝛿 180° − 𝛽

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Angle sum of a polygon – Method 1

  • Consider an 𝑜 -sided

polygon (𝑜 = 5 shown)

  • Divide the polygon into

𝑜 − 2 triangles by adding 𝑜 − 3 diagonals

  • By angle sum in each

triangle, the total angle sum is 𝑜 − 2 × 180°

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180° − 𝛾 180° − 𝛿 180° − 𝛽

Angle sum of a polygon – Method 2

  • Consider an 𝑜 -sided polygon

with angles 𝛽, 𝛾, 𝛿,…

  • Extend sides to form exterior

angles 180° − 𝛽 etc.

  • Shrink polygon to a single

point with 𝑜 exterior angles

  • By adjacent angles in a

revolution 180° − 𝛽 + 180° − 𝛾 + 180° − 𝛿 + ⋯ = 360°

  • Hence

𝛽 + 𝛾 + 𝛿 + ⋯ = 𝑜 − 2 × 180°

𝛽 𝛾 𝛿

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Properties of a regular pentagram

  • Inscribe inside a regular pentagon
  • Internal angles of pentagon are

3 × 180° ÷ 5 = 108°

  • By isosceles triangles, angle at

each vertex of regular pentagram is 36°, with sum 180°

  • GeoGebra link:

https://tinyurl.com/CMA2018pentagram 108° 72° 36° 36° 36° 36° 36° 36° 36° 108°

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Properties of a regular pentagram

  • Inscribe inside a regular pentagon
  • Internal angles of pentagon are

3 × 180° ÷ 5 = 108°

  • By isosceles triangles, angle at

each vertex of regular pentagram is 36°, with sum 180°

  • By similar triangles, the ratio of

diagonal and side lengths is the Golden Ratio 𝜚 =

1+ 5 2 , which is

related to Fibonacci etc.

𝝔 𝟐

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Angle sum of a cyclic pentagram

  • Consider a cyclic pentagram with

angles 𝛽, 𝛾, 𝛿, 𝜀, 𝜁 at the vertices

  • Construct radii through centre 𝑃
  • pposite angle 𝛽
  • By circle geometry theorem, the

associated angle at 𝑃 is 2𝛽

𝑃 𝛽 𝛾 𝛿 𝜀 𝜁 2𝛽

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Angle sum of a cyclic pentagram

  • Consider a cyclic pentagram with

angles 𝛽, 𝛾, 𝛿, 𝜀, 𝜁 at the vertices

  • Construct radii through centre 𝑃
  • pposite angle 𝛽
  • By circle geometry theorem, the

associated angle at 𝑃 is 2𝛽

  • Similarly for all other angles
  • By adjacent angles in a revolution

2𝛽 + 2𝛾 + 2𝛿 + 2𝜀 + 2𝜁 = 360°

  • Hence 𝛽 + 𝛾 + 𝛿 + 𝜀 + 𝜁 = 180°

𝛽 𝛾 𝛿 𝜀 𝜁 2𝛽 2𝛿 2𝜁 2𝛾 2𝜀

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𝜄

Angle sum of any pentagram

  • Consider any pentagram with

angles 𝛽, 𝛾, 𝛿, 𝜀, 𝜁 at the vertices

  • Consider triangle with angles 𝛽, 𝛾

and third angle 𝜄 = 180° − 𝛽 − 𝛾

𝛽 𝛾 𝛿 𝜀 𝜁

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𝜄

∙ ∙∙ ∙

Angle sum of any pentagram

  • Consider any pentagram with

angles 𝛽, 𝛾, 𝛿, 𝜀, 𝜁 at the vertices

  • Consider triangle with angles 𝛽, 𝛾

and third angle 𝜄 = 180° − 𝛽 − 𝛾

  • Similarly for the four other angles
  • f the internal pentagon
  • By angle sum of a pentagon

180° − 𝛽 − 𝛾 + 180° − 𝛾 − 𝛿 + 180° − 𝛿 − 𝜀 + 180° − 𝜀 − 𝜁 + 180° − 𝜁 − 𝛽 = 540°

  • Hence… 𝛽 + 𝛾 + 𝛿 + 𝜀 + 𝜁 = 180°

𝛽 𝛾 𝛿 𝜀 𝜁

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Constructing regular stars

  • Choose a number of vertices 𝑜
  • Label 𝑜 vertices around a circle

0, 1, 2, … , 𝑜 − 1

  • Choose a skipping number 𝑢
  • For each 𝑙 = 0, 1, 2, … , 𝑜 − 1, join

vertex 𝑙 to vertex 𝑙 + 𝑢 (mod 𝑜)

  • Call this star 𝑇(𝑜, 𝑢)
  • The example shown is 𝑇 5,2
  • Reversing the order, it is also 𝑇 5,3

1 4 2 3 𝑜 = 5 𝑢 = 2

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Constructing regular stars

1 4 2 3 1 4 2 3 𝑇 5,2 = 𝑇(5,3) 𝑇 5,1

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Constructing regular stars

1 4 2 3 1 4 2 3 1 4 2 3 𝑇 5,2 = 𝑇(5,3) 𝑇 5,1 = 𝑇(5,4) 𝑇 5,0 = 𝑇(5,5) 1 𝑇 2,1 1 2 𝑇 3,1 1 3 2 𝑇 4,1 1 3 2 𝑇 4,2

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Constructing regular stars

1 4 2 3 1 4 2 3 1 4 2 3 𝑇 5,2 = 𝑇(5,3) 𝑇 5,1 = 𝑇(5,4) 𝑇 5,0 = 𝑇(5,5) 1 𝑇 2,1 1 2 𝑇 3,1 1 3 2 𝑇 4,1 1 3 2 𝑇 4,2 = 2 × S(2,1)

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Constructing regular stars

𝑇(6,1) 𝑇(7,1) 𝑇 6,2 = 2 × 𝑇(3,1) 𝑇(7,2) 𝑇 6,3 = 3 × 𝑇(2,1) 𝑇(7,3)

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Constructing regular stars

𝑇 9,2 𝑇 9,3 = 3 × 𝑇(3,1) 𝑇 9,4 𝑇 8,2 = 2 × 𝑇 4,1 𝑇 8,3 𝑇 8,4 = 4 × 𝑇(2,1)

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Stars and number theory

  • 𝑇 𝑜, 𝑢 = 𝑇(𝑜, 𝑜 − 𝑢), so we can assume 0 ≤ 𝑢 ≤

𝑜 2

  • 𝑇 𝑜, 1 is an 𝑜-sided polygon
  • If 𝑒 = gcd 𝑜, 𝑢 , then 𝑇 𝑜, 𝑢 is 𝑒 rotated copies of 𝑇

𝑜 𝑒 , 𝑢 𝑒

  • Therefore 𝑇 𝑜, 𝑢 is connected if and only if gcd 𝑜, 𝑢 = 1
  • For 𝑜 ≥ 3, the number of different connected stars with

𝑜 vertices is

𝜒 𝑜 2 , where 𝜒 is the Euler-totient function

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Stars and complex numbers

  • It is well known that if 𝜕 = cis

2𝜌 𝑜

is the principal 𝑜th root of unity, the powers 1, 𝜕, 𝜕2, 𝜕3, … , 𝜕𝑜−1 form a regular 𝑜-gon in the complex plane

  • In general, if 𝑨 = 𝜕𝑢 is another 𝑜th root of unity, then

the powers 1, 𝑨, 𝑨2, 𝑨3, … , 𝑨𝑜−1 form 𝑇(𝑜, 𝑢)

  • This follows from index laws and the fact that 𝑨𝑜 = 1

so indices can be reduced modulo 𝑜

  • Adjusting the magnitude of 𝑨 introduces spiralling
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Angle sum of any pentagram

  • Consider any pentagram with angles

𝛽, 𝛾, 𝛿, 𝜀, 𝜁 at the vertices

  • Extend sides to form exterior angles

180° − 𝛽 etc.

  • Shrink pentagram to a single point

with 𝑜 exterior angles

𝛽 𝛾 𝛿 𝜀 𝜁

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SLIDE 45

Angle sum of any pentagram

  • Consider any pentagram with angles

𝛽, 𝛾, 𝛿, 𝜀, 𝜁 at the vertices

  • Extend sides to form exterior angles

180° − 𝛽 etc.

  • Shrink pentagram to a single point

with 𝑜 exterior angles

𝛽 𝛾 𝛿 𝜀 𝜁

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Angle sum of any pentagram

  • Consider any pentagram with angles

𝛽, 𝛾, 𝛿, 𝜀, 𝜁 at the vertices

  • Extend sides to form exterior angles

180° − 𝛽 etc.

  • Shrink pentagram to a single point

with 𝑜 exterior angles which account for 𝟑 full revolutions

  • Then

180° − 𝛽 + 180° − 𝛾 + 180° − 𝛿 + 180° − 𝜀 + 180° − 𝜁 = 2 × 360°

  • Hence

𝛽 + 𝛾 + 𝛿 + 𝜀 + 𝜁 = 180°

𝛽 𝛾 𝛿 𝜀 𝜁 𝑇(5,2) 𝑢 = 2

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Angle sum of any star

  • Consider the star 𝑇(𝑜, 𝑢) with angles 𝛽1, 𝛽1, …, 𝛽𝑜 at the vertices

(note that we do not need to assume they are equal!)

  • The exterior angle at vertex 𝑙 is 180° − 𝛽𝑙
  • Since 𝑢 vertices are skipped at each step, it takes 𝑢 revolutions

to trace around the full star

  • Hence the sum of exterior angles is
  • Therefore the sum of angles at the vertices is (𝑜 − 2𝑢) × 180°

𝑢 × 360° = 𝑜 × 180° − ෍

𝑙=1 𝑜

𝛽𝑙 = ෍

𝑙=1 𝑜

180° − 𝛽𝑙 2𝑢 × 180° =

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  • Theorem: If 1 ≤ 𝑢 ≤

𝑜 2 , the angle sum of the,

possibly irregular, star 𝑇(𝑜, 𝑢) is 𝑜 − 2𝑢 × 180°.

  • Corollary 1: Letting 𝑢 = 1, the angle sum of an

𝑜-sided polygon is (𝑜 − 2) × 180°.

  • Corollary 2: If 1 ≤ 𝑢 ≤

𝑜 2 , the regular star 𝑇(𝑜, 𝑢)

has angles of 1 −

2𝑢 𝑜

× 180° at each vertex.

Angle sum = (𝑜 − 2𝑢) × 180°

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Angle sum = (𝑜 − 2𝑢) × 180°

𝑇 6,2 σ 𝛽 = 360° 𝑇 6,3 σ 𝛽 = 0° 𝑇 7,2 σ 𝛽 = 540° 𝑇 8,3 σ 𝛽 = 360° 𝑇 9,4 σ 𝛽 = 180°

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Angle sum = (𝑜 − 2𝑢) × 180°

  • Example: Which regular connected star has an

angle of 54° at each vertex?

  • Solution:

1 −

2𝑢 𝑜

× 180° = 54° 1 −

2𝑢 𝑜 = 54 180 = 3 10 2𝑢 𝑜 = 7 10

so

𝑢 𝑜 = 7 20

gcd 𝑢, 𝑜 = 1, since the star is connected, hence 𝑜 = 20 and 𝑢 = 7, so the star is 𝑇(20,7)

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Turtle programming with Python

  • Free online Python compiler:

https://trinket.io/library/trinkets/create?lang=python

  • Example: Plot regular 𝑇 20,7 , with exterior angles of

180° − 54° = 126° from turtle import * for k in range(20): forward(100) left(126)

must keep indents here to ensure these two commands are part of the ‘for loop’

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Turtle programming with Python

  • Example: Plot any regular connected 𝑇 𝑜, 𝑢 , gcd 𝑜, 𝑢 = 1

from turtle import * n=20 # number of vertices t=7 # skipping number s=100 # side length e=360*t/n # exterior angle at vertices for k in range(n): forward(s) left(e)

  • For gcd 𝑜, 𝑢 ≠ 1 see https://tinyurl.com/CMA2018stars
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Other Turtle features

right(…) # turn right by given angle back(…) # move backwards by given distance dot(…) # draw dot of given size penup() # lift pen to move without drawing pendown() # resume drawing pensize(…) # any decimal value greater than 0, default is 1 pencolor(…) # e.g. ’red’, ’blue’ etc., including quotes begin_fill() # turn on colour filling, must be paired with… end_fill() # turn off colour filling hideturtle() # hides arrowhead while tracing speed(…) # 1 = slowest, 100 = fastest tracer(0,0) # turns off tracing, plots final output only update() # must put this at the end if using tracer(0,0)

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References

  • Cut the Knot. 2018. Golden Ratio in Geometry.

https://www.cut-the-knot.org/do_you_know/GoldenRatio.shtml

  • Brilliant.org. 2018. Roots of unity.

https://brilliant.org/wiki/roots-of-unity/

  • GeoGebra. 2018. GeoGebra Geometry.

https://www.geogebra.org/geometry

  • The Python Software Foundation. 2018. Turtle graphics for Tk.

https://docs.python.org/2/library/turtle.html

  • Trinket. 2018. Available Python Modules.

https://trinket.io/docs/python

  • Wikipedia. 2018. Euler's totient function.

https://en.wikipedia.org/wiki/Euler%27s_totient_function

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Image sources

  • Aruba flag:

https://en.wikipedia.org/wiki/Flag_of_Aruba

  • Australia flag:

https://en.wikipedia.org/wiki/Commonwealth_Star

  • Azerbaijan flag:

https://en.wikipedia.org/wiki/Flag_of_Azerbaijan

  • Baha’i symbol:

https://www.artfire.com/ext/shop/

  • Ethiopia flag:

https://en.wikipedia.org/wiki/Flag_of_Ethiopia

  • Iranian tile:

https://www.pinterest.co.uk/NtlMuseumsScot/

  • Israel flag:

https://en.wikipedia.org/wiki/Flag_of_Israel

  • Malaysia flag:

https://en.wikipedia.org/wiki/Flag_of_Malaysia

  • Marshall Islands flag:

https://en.wikipedia.org/wiki/Flag_of_the_Marshall_Islands

  • Morocco flag:

https://en.wikipedia.org/wiki/Flag_of_Morocco

  • Nauru flag:

https://en.wikipedia.org/wiki/Flag_of_Aruba

  • Saint Kitts and Nevis flag: https://en.wikipedia.org/wiki/Flag_of_Saint_Kitts_and_Nevis
  • Senegal flag:

https://en.wikipedia.org/wiki/Flag_of_Senegal

  • Slipknot logos:

http://www.fanpop.com/clubs/metal-gods/

  • Somalia flag:

https://en.wikipedia.org/wiki/Flag_of_Somalia

  • Star background:

https://gifer.com/en/9Y9F

  • Suffolk County badge:

http://socialistcurrents.org/?p=2173

  • Timor-Leste flag:

https://en.wikipedia.org/wiki/Flag_of_East_Timor