Download 34 Sept 2016 - U3A Site Builder

Steyning & District U3A
Discovering Mathematics
Session 34
Two Trig. Problems
& Symmetry
A Simple Trig. Problem
x
θ
20
y
The diameters of the 2 circles are;
10 & 14 cm resp. & their centres are
located on the same horizontal line
20 cm apart.
What is the angle formed between
the common tangent & the
horizontal?
The radii of the 2 circles are each at right angles to the common tangent & the angle
θ is common to both triangles. Therefore the triangles are similar.
Thus the ratio radius/hypotenuse will be equal for each.
ie 5/x = 7/y or 5y =7x & y = 7x/5
But x + y = 20
therefore; x + 7x/5 = 20 or 5x + 7x = 20*5, 12x = 100 or x = 25/3
and y = 20 – 25/3 = (60 - 25)/3 = 35/3
Now, looking at the large triangle we can see that;
Sin θ = side opp./hypotenuse = 7/y = 7*3/35 = 3/5 = 0.6 or θ = 36.90
Check the same result for the small triangle; Sinθ = 5/x = 15/25 = 3/5.
Radius of a Wheel
A wheel is at rest against a step measuring 10 x 5
cm
The extended lines from the bottom and back of the
step are tangential to the wheel perimeter.
What is the wheel radius?
r
x
5
y
x = (r – 10) & y = (r -5) and r2 = x2 + y2
Substituting; r2 = (r-10)2 + (r – 5)2
2 = (r2 – 20r + 100) + (r2 – 10r +25)
Expanding;
r
10
And r2 = 2r2 – 30r + 125 or r2 – 30r +125 = 0
Factorizing; (r – 25)(r – 5) = 0,
therefore; r = 25 or r = 5
However, if r = 5, then x = 5 – 10 which is negative and impossible.
Therefore the only feasible solution is; r = 25
Mathematical Symmetry
An object is said to be symmetric if its shape appears to be unchanged after it has
been moved or transformed.
The 3 basic types of transformations are; reflections, rotations and translations.
Reflection is defined as a mirror symmetry about a line for a 2 dimensional object, or
about a plane in the case of a 3 dimensional object. The lengths or angles of the
object remain unchanged.
The line or plane is defined as the axis of symmetry.
Mathematical Symmetry (cont.)
Rotations are defined as instances where an object is rotated around an axis, or
for a 3 dimensional object, is moved around a plane.
A Translation occurs when an object is simply moved in an unchanged state in a defined
direction or directions.
Mathematical Symmetry (Cont. 2)
Combinations of 2 or even 3 of these transformations can also be carried out.
This is an example of a combined reflection and translation.
Three Dimensional Symmetry
The mathematicians off ancient Greece were fascinated with symmetry and
particularly with three dimensional symmetry.
A polygon is a 2 dimensional object drawn with straight lines, such as a triangle or a
rectangle, etc. Regular polygons have the additional feature in that the lengths of all
sides are equal, eg. an equilateral triangle, a square, pentagon, hexagon, etc.
For each number of sides there is only one regular polygon.
This simple concept is not repeated for 3 dimensional symmetrical shapes which are
known as polyhedra, such as the cube. The cube is built from 6 surfaces which
themselves are regular polygons, ie squares.
The Greek geometers were well aware of 2 other polyhedra, the tetrahedron, which
is a pyramid with a triangular base and the octahedron, which has 8 faces and is
effectively 2 square based pyramids joined at their square faces.
It was Theaetetus (417 – 369 BC) who developed a theorem to demonstrate that
there are only 2 additional regular polhedra, the dodecahedron, having 12
pentagonal faces and the icosahedron, having 20 equilateral triangular faces.
Theaetetus described the 5 regular polyhedra; ‘the Platonic Solids’ in honour of his
close personal friend, Plato.
Theaetetus’ Theorem
This is a beautifully elegant proof of the theorem;
•
•
•
•
•
•
•
•
•
Each vertex must include at least 3 faces.
The total included angle of all the faces at each vertex must be less than 3600.
The angles of each face must therefore be less than 360/3 ie < 120o.
Regular pentagons with 6 sides or more have face angles of 120o or more.
Thus only triangular, square or pentagonal faces can be utilised to construct a
regular polyhedron.
A triangular polygon with 60o can therefore be utilized for 3,4 or 5 faces at each
vertex, totalling less than 360o. These produce the tetrahedron, the octahedron
and the icosahedron, resp..
Each vertex using square faces has an angle of 90o. Thus only a 3 face vertex
meets the < 360o criterium, producing a cube.
The angle of a regular pentagon is 108o, so only one vertex of three faces is
possible. This is the dodecahedron.
The conclusion therefore is that there are only 5 polyhedra which meet these
criteria.
Three Dimensional Symmetry (cont. 1)
The Platonic Solids
Tetrahedron
(4 equilateral faces)
Cube
(6 square faces)
Dodecahedron
(12 pentagonal faces)
Octahedron
(8 equilateral faces)
Icosahedron
(20 equilateral faces)
Images from Wikimedia Commons
Three Dimensional Symmetry (Cont. 2)
Theaetetus had assumed that the faces & edges of regular polyhedra could not
self-intersect, but Kepler & later, Poinsot, each demonstrated that 2 more regular
polyhedra were possible, making a total of 9 regular polyhedra.
Small Stellated
Dodecahedron
Great Dodecahedron
Great Stellated
Dodecahedron
Great Icosahedron
Images from Wikimedia Commons
Mathematical Symmetry
Symmetry by Prof. Marcus du Sautoy
That’s it Folks
Suggestions for future topics?