Spirograph (toy)

Original spirograph box

Spirograph clone in use

different patterns

Animation of the formation of a hypotrochoid

Animation of the formation of an epitrochoid

Spirograph is a geometric toy that can be used to draw various patterns or mathematical curves .

functionality

The spirograph consists of several, mostly round, thin gears made of plastic disks. First, a sheet of paper is placed on a cardboard. Then, depending on the design, a larger toothed plastic ring or an internally toothed hole template is attached to it (in the original with needles). Inside (or on the exterior) of the ring gear of the gears is applied. The teeth interlock like a rack . There are holes in the gears at different distances through which the tip of a writing instrument is inserted. Here you have to z. B. write a circle with a ballpoint pen in the toothed disk.

By using several colored ballpoint pens or pens in different holes, you get different geometric figures, so-called hypocycloids and epicycloids .

The inventor was Denys Fisher in 1965 , who presented the toy for the first time at the Nuremberg Toy Fair . There are many different versions, also with a motor. The spirograph won the 1967 Toy of the Year Award .

But even before Denys Fisher there were at least two inventors who had spiral drafters patented: Bruno Abdank-Abakanowicz in 1885 and Ernst Barthel in 1933.

Mathematical description

The hypotrochoid is formed when a small circle is rolled inside a large circle and the epitrochoid when a small circle is rolled outside a large circle. The animations on the right make this clear. With the parameters listed below, the parameterization of epi- and hypotrochoid can be specified as follows:

${\ displaystyle R}$ : Radius of the fixed circle (internally toothed hole template for hypo- and externally toothed epitrochoid)
${\ displaystyle r}$ : Radius of the moving circle (gear)
${\ displaystyle a}$ : Radius of the pin position in relation to the gear center point
${\ displaystyle \ nu = {\ begin {cases} 1 &: {\ text {Epitrochoid}} \\ - 1 &: {\ text {Hypotrochoid}} \ end {cases}}}$ : Selection parameters

{\ displaystyle {\ varvec {x}} (t) = (R + \ nu \ cdot r) \ cdot {\ begin {pmatrix} \ cos (t) \\\ sin (t) \ end {pmatrix}} - a \ cdot {\ begin {pmatrix} \ nu \ cos \ left ({\ frac {R + \ nu \ cdot r} {r}} \ cdot t \ right) \\\ sin \ left ({\ frac {R + \ nu \ cdot r} {r}} \ cdot t \ right) \ end {pmatrix}}}

If the parameter is selected, the curves are referred to as epi- or hypocycloids. These represent, so to speak, a special case of the epi- and hypotrochoid. However, such a configuration is not possible with classic Spriograph toys. The pen is always in one position here . ${\ displaystyle a = r}$ ${\ displaystyle a <r}$

The following illustration is used to describe periodic figures . Are and prime , then stands for the number of revolutions of the gear until the curve is closed. The common period with respect to the parameter is consequently . ${\ displaystyle n, m \ in \ mathbb {N}}$ ${\ displaystyle m}$ ${\ displaystyle n}$ ${\ displaystyle n}$ ${\ displaystyle t}$ ${\ displaystyle T = 2 \ pi n}$

{\ displaystyle {\ boldsymbol {x}} (t) = {\ frac {Rm} {m- \ nu n}} \ cdot {\ begin {pmatrix} \ cos (t) \\\ sin (t) \ end {pmatrix}} - a \ cdot {\ begin {pmatrix} \ nu \ cos ({\ frac {m} {n}} \ cdot t) \\\ sin ({\ frac {m} {n}} \ cdot t) \ end {pmatrix}} \ quad {\ text {with}} \ quad {\ frac {R} {r}} = {\ frac {m} {n}} - \ nu}

Special hypotrochoid ( ), which run through the center of the circle with a radius , result from the selection of . In this case the ratio continues to apply . This is the case with the hypotrochoid shown in the adjacent animation. If, on the other hand , you choose, star-shaped curves are obtained, the tips of which lie on the circle with a radius . Furthermore, in this case, the number of star peaks results and the ratio applies . ${\ displaystyle \ nu = -1}$ ${\ displaystyle R}$ ${\ displaystyle a = {\ frac {R \ cdot m} {m + n}}}$ ${\ displaystyle {\ frac {a} {r}} = {\ frac {m} {n}}}$ ${\ displaystyle a = {\ frac {R \ cdot n} {m + n}}}$ ${\ displaystyle R}$ ${\ displaystyle k = n + m}$ ${\ displaystyle {\ frac {a} {r}} = 1}$