# The maths behind Spirograph

# The maths behind Spirograph

### Introduction:

Spirograph is a popular toy that has fascinated people for generations. It consists of a set of plastic gears, a ring, and a set of pens or pencils. By moving the pens or pencils along the ring's edges as it rotates inside the gears, users can create intricate and beautiful designs. The mathematical principles of hypotrochoids and epitrochoids underlie the creation of these designs. In this blog, we will explore these principles and how they relate to the maths of Spirograph.

Hypotrochoids:

A hypotrochoid is a curve that is created by the movement of a point on a small circle that rolls around the inside of a larger circle. The movement of the point is determined by the size of the small circle, the size of the large circle, and the distance between the center of the small circle and the center of the large circle.

In Spirograph, the small circle is represented by the pen or pencil, and the large circle is represented by the ring. As the ring rotates inside the gears, the pen or pencil moves along the edge of the ring, creating a hypotrochoid.

The equation for a hypotrochoid is given by:

x = (R-r) * cos(t) + d * cos(((R-r)/r) * t)

y = (R-r) * sin(t) - d * sin(((R-r)/r) * t)

Where:

R is the radius of the large circle (represented by the ring in Spirograph)

r is the radius of the small circle (represented by the pen or pencil in Spirograph)

d is the distance between the center of the small circle and the center of the large circle (represented by the distance between the pen or pencil and the ring in Spirograph)

t is the angle of rotation

By varying the values of R, r, and d, users can create a wide range of hypotrochoid patterns in Spirograph.

Epitrochoids:

An epitrochoid is a curve that is created by the movement of a point on a small circle that rolls around the outside of a larger circle. The movement of the point is determined by the size of the small circle, the size of the large circle, and the distance between the center of the small circle and the center of the large circle.

In Spirograph, an epitrochoid is created when the pen or pencil is placed outside of the ring and moves along the edge of the ring as it rotates inside the gears.

The equation for an epitrochoid is given by:

x = (R+r) * cos(t) - d * cos(((R+r)/r) * t)

y = (R+r) * sin(t) - d * sin(((R+r)/r) * t)

Where:

R is the radius of the large circle (represented by the ring in Spirograph)

r is the radius of the small circle (represented by the pen or pencil in Spirograph)

d is the distance between the center of the small circle and the center of the large circle (represented by the distance between the pen or pencil and the ring in Spirograph)

t is the angle of rotation

By varying the values of R, r, and d, users can create a wide range of epitrochoid patterns in Spirograph.

Combining Hypotrochoids and Epitrochoids:

By combining hypotrochoids and epitrochoids, users can create even more intricate and beautiful patterns in Spirograph. This is done by using

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a combination of gears and rings with different sizes and shapes, which allows for more complex movements of the pen or pencil along the edge of the ring.

For example, by using a gear with a certain number of teeth and a ring with a certain number of holes, users can create a pattern that repeats after a certain number of rotations. This is because the movement of the pen or pencil is determined by the interaction of the teeth and holes, which create a set number of unique positions for the pen or pencil to move along the edge of the ring.

Spirograph History:

Spirograph was invented in the early 1960s by British engineer Denys Fisher. Fisher was inspired by the geometric art created by artist and mathematician Bruno Ernst, who used a similar method of creating hypotrochoids and epitrochoids with a set of gears and circles.

Fisher's Spirograph toy quickly became a sensation, winning the 1967 Toy of the Year Award in the United Kingdom and becoming a popular toy around the world. Over the years, Spirograph has undergone numerous adaptations and variations, including digital versions and larger sets with more gears and rings.

Conclusion:

Spirograph is a classic toy that uses the mathematical principles of hypotrochoids and epitrochoids to create intricate and beautiful designs. By varying the sizes and shapes of gears and rings, users can create a wide range of patterns that are impossible to draw by hand. The popularity of Spirograph has endured for decades, inspiring generations of artists and mathematicians to explore the beauty and complexity of mathematical curves.