Each of the six equally spaced points on this circle has been joined to a point that is two point aw

1
STARS

• Each of the six equally spaced points on this
  circle has been joined to a point that is two
  point away from it, in a clock-wise direction.
• The result can be called a 6, 2 figure.
• In general the result for n points each joined
  to the point p points away from it is called an
  n, p figure.
• Equally spaced points on a circle can be joined
  by chords in various ways.
• Investigate

6, 2
2
Breaking down the investigation
This investigation is a geometry and algebra
statement. Geometry is the expression of
relationships between points, lines, shapes and
in this case circles and chords are used. The
term in general used in the investigation
question imposed that geometry can be generalised
as like any other mathematical problems. The 6,
2 are the co-ordinates the points are n 6,
and the p represents the position 2.
n
n
p
n
p
In a clockwise direction
n
n
6, 2
n
3
Looking for a pattern

• The first pattern found involved drawing the
  circles with a compass and protractor measuring
  the points to be equally spaced.
• By joining up all the chords in a 6, 1 figure
  created a hexagon, 6, 2 a star with two
  triangles, 6, 3 figure a crossing over bicycle
  wheel. The 6, 5 figure a hexagon.

6, 1
6, 2
6, 3
6, 4
6, 5
4
What was found from this pattern

• The figures were drawn up to twelve to show a
  more in-depth look at the patterns. As the points
  increased the stars and shapes became intense
  forming many angles.
• These angles when counted summed up to the amount
  of co-ordinates.
• The shapes were still apparent even when the p
  was only one co-ordinate, a sequence was becoming
  apparent.

pentagon
square
hexagon
circle
triangle
5
What other issues can be identified

• Parallel lines are evident and angles. These
  being equal or vertically opposite. The
  corresponding angles are equal pairs. Therefore
  showing lines of symmetry and types of triangles
  and shapes that reflect each other.
• An overlapping square is seen, and two lines of
  symmetry are evident. Also angles of triangles
  are measured to 190 degrees that also make up a
  90 degree square.

8, 2
8, 5
6
overlapping figures
Why a clockwise direction? Figures that are in
continuous or anti-clockwise i.e triangle over a
triangle, are when n and p are devisable, or
even. Figures that start and finish at the same
point and involve every other point in the
process are when n and p are not equally
divisible or odd, creating figures that overlap.
(0.7)
2
5
4
3
even number
6
odd number
1
6, 10
7, 4
7
Number of regions
A reflective pattern occurs when counting the
regions.
When you get to the 6. 6 figure which the
co-ordinates are equal no chords can be joined,
and on either side of this co-ordinate a
reflective mirror pattern occurs. Therefore
predicting what will come next.
8
What can we predict?

• By substituting the n and p for figures, i.e.
  n3, and p4, you know that for each set of
  co-ordinates, shapes arise.
• By counting the regions, they form a pattern of
  reflection.
• Overlapping figures have parallel lines, angles
  and symmetry of shapes.
• Measuring accurately prediction when counting
  regions, and looking for patterns.
• Rectangles and other shapes have an axis of
  symmetry.

9
Number of intersections
Reflections in the pattern occur after working
out the number of intersections that cross.
5, 3
Five intersections crossed
10
Angles
6060120

• equilateral triangles
• quadrilaterals
• obtuse and reflex angles
• acute angles
• interior and exterior angles
• equal or corresponding angles
• parallel lines with angles that
  correspond.

60
60
60
60
180
60
Angles measured using a protractor Sum of a
triangle a b c 180 The exterior angle of a
triangle is a b 120
6, 2
11
Geometry and Algebra

• Topics that can be investigated include
• Shapes and their properties
• Parallel lines
• Types of angles
• Area and length of the shapes
• The formula for the area of a circle is
• radius x circumference r x 2?r ?r²
• 2
• Trigonometry to work out degrees of triangles.
• Algebra to work out the base, angle and point of
  intersections.

12
Geometry and Algebra
Geometry is the study of space and shape. Looking
at properties of the shape, working out the area,
length and volume. Algebra expressions are
frequently used in many problems, but in this
case geometry. These can be expressed as
formulas. As a result this investigation can
lead into many areas of mathematics. These are
only a few patterns, Ive investigated. There are
many mathematicians that work out formulas using
Euclids Theorem, the example on the right was
beyond me, but I am working on it!
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