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Introduction
Number Patterns
Pictorial Patterns
Tile Patterns
Magic Squares
Patterns in Statistical Experiments
Conclusions
Bibliography
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The Mystic Rose
![[Picture of the Mystic Rose]](mapicp2.jpg) To make a mystic rose pattern as shown on the right, equal intervals are marked off on the circumference of a circle and each point is joined to every other one.
We can calculate the number of lines that need to be drawn to finish the pattern by adding the series "n, n-1, n-2, n-3, n-4, ..., 0", where n is the number of marks on the circumference of the circle. This sum is easily calculated by averaging the first and last terms of the sum and multiplying that result by the number of terms in the sum.
An useful application of the Mystic Rose is for sporting competitions involving a league structure, where teams play each other once. If each node on the circumference is represented by a team in the league, we can see how many games that need to be played by adding the series shown above. Also, if we track a season by drawing lines from team to team as every game is played, we can analyse which games are left. Finally, if the number of teams in the league increases or decreases, we can easily recalculate the number of matches in a season by again using the series shown above.
Pascal's Theorem
![[Picture of Pascal's Triangle]](mapicp3.jpg) Pascal's Theorem says that if any hexagon is drawn inside a conic section (i.e. a circle, parabola, ellipse or hyperbola), then the points of intersection of its three pairs of lines, produced, will lie on a straight line. Note here that the envelope of the "ellipse" drawn is an approximation constructed from four rectangular parabolas.
The fact that intersections occur on a single straight line is used in practical applications. One example of this is the design of reflectors for electric fires. They are parabolic in shape because all rays of light or heat emanating from the focus and striking the reflector will be reflected in a direction parallel to the axis.
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