The Archimedean Spiral

Abstract

Archimedean spirals are described, as well as spirals that look quite similar optically and can serve as approximations of the Archimedean spiral.

Archimedean spirals are described, as well as spirals that look quite similar optically and can serve as approximations of the Archimedean spiral.

3.1 Linear Radius Function

In polar coordinates, an Archimedean spiral is represented as:

$$ r\left( t \right) = at + b $$

(3.1)

So, a linear function is “wound up”. Casual formulation: the more rotation, the more distance.

Figure 3.1a initially shows the radius function for the example

$$ r\left( t \right) = \frac{1}{2\pi }t,\;0 \le t \le 6\pi $$

(3.2)

Fig. 3.1

Archimedean spiral

in a Cartesian $t$-$r$-coordinate system. The radius $r$ grows uniformly with the rotation angle $t$. Figure 3.1b shows the corresponding Archimedean spiral.

It starts for the rotation angle $t = 0$ at the origin. For the rotation angle $t = 2\pi$ is $r\left( {2\pi } \right) = 1$, the spiral intersects the $x$-axis at the point 1. Further, $r\left( {4\pi } \right) = 2$ and $r\left( {6\pi } \right) = 3$ with the corresponding intersection points on the $x$-axis. The spiral has a total of 3 turns.

3.2 Interstice

The spiral (Fig. 3.1) intersects the positive $x$ axis at the points 0, 1, 2, 3. Therefore, the horizontal distance between two spiral turns is always the same on the positive x-axis, namely one.

However, the positive $x$ axis does not intersect the spiral at a right angle, but at an angle. In addition, the angle of intersection to the $x$ axis varies, as can be seen from the blue tangents drawn in Fig. 3.2. However, the angle of intersection increasingly approaches a right angle towards the outside.

Fig. 3.2

How thick is the spiral?

The two inserted colored circles both have a diameter of 1. The yellow circle near the center cannot be fitted between two spiral turns. The distance between the two spiral turns is significantly less than 1. The situation is less clear with the green circle further out. But it also does not fit. This can be seen in an exact drawing with dynamic geometry software by zooming in.

A circle with a diameter slightly less than 1 can be inserted far enough outside between two spiral turns. However, as it approaches the center, it gets stuck somewhere. The Archimedean spiral is not “equally thick” everywhere, although this seems to be the case at first glance.

However, there are also spirals that are actually the same thickness everywhere (Sects. 3.4, 3.5, 3.6 and 3.7).

3.3 The Parabola Comes into Play

A family of uniformly twisted Archimedean spirals (Fig. 3.3a) is cut by a horizontal ray emanating from the center at regular intervals. The tangents at these intersection points are not parallel (Fig. 3.3b). They become steeper towards the outside.

Fig. 3.3

Spiral family with tangents

The envelope of these tangents is a parabola (Fig. 3.4a).

Fig. 3.4

Parabola and focus

The normals to the spirals at the intersection points pass through a common point (Fig. 3.4b). This is the focus of the parabola. Proof by calculation. The animation (Fig. 3.5) illustrates the situation.

Fig. 3.5

Tangent (▶ https://doi.org/10.1007/000-br6)

3.4 Offset Circles

A concentric set of circles with equal distances is cut horizontally (Fig. 3.6a). Subsequently, the upper half is shifted one unit to the right (Fig. 3.6b). By suitable recoloring, two intertwining spirals are formed (Fig. 3.6c). The spirals are not true Archimedean spirals, but are composed of semicircles.

Fig. 3.6

Offset semicircles

At any point in the figure, circular disks of the same diameter can be fitted (Fig. 3.7).

Fig. 3.7

Fitting circular disks

The semicircles of a spiral smoothly transition into each other at the transition points of the semicircles. However, since their radii are unequal, a curvature jump occurs at each transition point. Nevertheless, circular disks can also be fitted there.

The two intertwining spirals can be combined into a band ornament (Fig. 3.8). This results in interlinked double spirals.

Fig. 3.8

Band ornament

What happens if the upper half is shifted two or three units to the right?—With a shift of two units, three colors are needed (Figs. 3.9 and 3.10).

Fig. 3.9

Offset by two units

Fig. 3.10

Band ornament with three colors

A shift by three units simplifies the color choice (Fig. 3.11).

Fig. 3.11

Offset by three units

The corresponding band ornament is quite simple (Fig. 3.12). They are interlinked S-curves.

Fig. 3.12

Band ornament from S-curves

Can the set of circles also be divided into thirds?—Fig. 3.13 shows an example (with a small equilateral triangle in the center). The curvature jumps are now clearly visible, especially in the center of the figure.

Fig. 3.13

Dividing the set of circles into thirds

Of course, this can be done accordingly with quarters, fifths, sixths, and so on.

Instead of a band ornament, dividing by three results in a surface ornament with a hexagonal structure. So, a baroque honeycomb pattern (Fig. 3.14).

Fig. 3.14

Hexagonal structure

As expected, dividing by four results in a square structure (Fig. 3.15). The image appears to be skewed, but it is correctly positioned. The small white squares in the spiral centers are parallel to the page margins. The skewed impression is due to the offset of the spiral parts.

Fig. 3.15

Square structure

Dividing by five presents geometric problems when joining the spirals. The figures do not close (Fig. 3.16a). The reason is that a regular surface ornament cannot contain five-part rotational symmetry. Therefore, a floor cannot be tiled with regular pentagons.

Fig. 3.16

Problems with dividing by five and six

When dividing by six, the figure closes in terms of lines, but the colors interfere (Fig. 3.16b). The problem of color collision can be solved with only three cyclically distributed colors (Fig. 3.17a).

Fig. 3.17

Dividing by six with three colors

Topologically, a simple triangle grid is created (Fig. 3.17b). The nodes are regular hexagons that have been left out.

At the nodes of Fig. 3.17b, you can apply an Allen key (hex key) and turn these keys synchronously to accomplish the unwinding and winding. This takes you from Fig. 3.17b back to 3.17a.

3.5 Coloring

In the examples of Figs. 3.6 to 3.17, the circle lines were colored. However, the situation can also be reversed: for given black circle lines (Fig. 3.18), one seeks to color the spaces in between. How many colors are needed?

Fig. 3.18

Coloring?

The two examples of Fig. 3.18 look almost the same. The difference only becomes clear when coloring. The example of Fig. 3.18a can be colored with two colors (Fig. 3.19a), but not with more than two colors.

Fig. 3.19

Number of colors

Figure 3.18b only allows one color (Fig. 3.19b), at least if the ornament is thought to be infinitely long.

With the phrase “thinking infinitely long,” the question of open ends is circumvented, where one could possibly enter with another color.

To avoid this linguistic caper, one can design the situation in a circular shape (Fig. 3.20). After all, a circle has no ends.

Fig. 3.20

Circular band ornaments

3.6 Unwinding and Winding

A string is attached to a vertical square rod and wound up. When subsequently unwinding, the outer end of the string describes a spiral. Figure 3.21 shows the view from above. From the square rod, only the cross-section, a square, is visible. For the following calculations, the side of the square is set to $1$.

Fig. 3.21

Unwinding

The spiral is composed of quarter-circle arcs. For example, with two turns, there are eight quarter-circle arcs with the successive radii 1, 2, 3, …, 8. The spiral length $s$ can therefore be easily calculated:

$$ s = \frac{\pi }{2}\left( {1 + 2 + \cdots + 8} \right) = 18\pi $$

(3.3)

With $n$ quarter-circle arcs, the spiral length $s$ is correspondingly:

$$ s = \frac{\pi }{2}\left( {1 + 2 + \cdots + n} \right) = \frac{\pi }{2} \frac{{n\left( {n + 1} \right)}}{2} $$

(3.4)

The spiral length $s$ thus grows quadratically with $n$.

The distance between two spiral turns is 4, which is the circumference of the square. The spiral is uniformly thick everywhere.

At the transition points of two successive quarter-circle arcs, there is a radius jump and thus a curvature jump.

The square can of course be replaced by another polygon. It does not need to be regular (Fig. 3.22).

Fig. 3.22

Triangle as starting figure

The distance between two spiral turns is the circumference of the wrapped figure.

3.7 Circle Evolute

If the square in Fig. 3.21 is replaced by a circle (3.23), a circle involute is created. The word comes from the Latin “evolvere” (to develop, unwind). The string is now wound around a cylinder. This reminds us of the caricature of the biting dog, whose long leash is tied to a tree trunk. A boy wants to tease him and runs around the tree at a safe distance. The dog tries to chase him and winds up his leash more and more. That’s how it can go.

Fig. 3.23

Circle evolute

The distance between two spiral passes is equal to the circumference of the circle. There are now no more jumps in curvature.

All circle involutes have the same shape. They can only differ in their size. Because the circle and the starting point of two circle involutes can be superimposed by a similarity transformation. Similarity transformations are translations, rotations, scalings (zooming) and combinations thereof. As soon as the circle and the starting point are mapped onto each other, the involutes also coincide. There are other figures in geometry that are all similar to each other, such as circle, square, equilateral triangle, parabola. The clothoid also fits into this (Chap. 6).

3.7.1 Cartography

Evolutes also result from the transfer of geographical latitude from the globe to the flat map (Fig. 3.24).

Fig. 3.24

From the globe to the flat map

To do this, a string is attached to the equator and stretched on the globe along a meridian to the desired geographical latitude. The string is then unwound until it stands vertically. This results in the corresponding geographical latitude on the flat map. Alternatively, the outline of the sphere can simply be rolled off on the left vertical edge of the map until the corresponding geographical latitude arrives there. The path curve of the outline point of the corresponding geographical latitude is a circle involute.

In the flat map, the meridians and the equator are reproduced without distortion in terms of length. The other latitude circles, however, are distorted, and this varies depending on the geographical latitude.

3.7.2 Gears

Circle involutes are used in mechanical engineering in the shaping of the teeth of gears (red in Fig. 3.25). The common tangent of the wheel circles can be interpreted as the unwinding string. The contact points of the teeth lie on the common tangent. The use of circle involutes ensures that the rotary motion is transferred evenly from one gear to the other.

Fig. 3.25

Gears

In the context of gears, circle involutes were already studied by Leonhard Euler (1701–1783).

3.7.3 Optical Effects

A circle involute can be equipped with congruent squares or rectangles due to the constant track width.

In Fig. 3.26a there are 101 numbered squares, the squares with even numbers are colored red, the others blue. In the center it looks a bit messy, but nicely lined up at the edge. Fig. 3.26b has 1001 squares without entered numbers, but with the alternating red-blue coloring. Optical effects with strange internal structures are created.

Fig. 3.26

Squares

In Fig. 3.27, rectangles with a side ratio of 3:1 are integrated into the circle involute. The rectangle arrangements are identical in both figures. The alternating red-blue coloring, however, makes patterns visible that would not be seen without coloring.

Fig. 3.27

Rectangles

In Fig. 3.28, the aspect ratio of the rectangles is slightly changed to $\pi {:}1$. The figures are now simple and clear.

Fig. 3.28

Circle number π

The background is as follows. After one revolution, the distance from the center is one rectangle width larger, the arc length is therefore like the circle by $2\pi$ larger, so just two rectangle lengths. With alternating coloring, after half a revolution two rectangles of different color and after a full revolution two rectangles of the same color come to lie on top of each other. After a quarter revolution, a shift by half a rectangle length occurs as in a brick masonry.

When modulating a sine curve onto a circle involute, a small change in the wavelength is sufficient to create a radial structure (Fig. 3.29a) or a slightly spiral structure (Fig. 3.29b) in the overall image.

Fig. 3.29

Sine curves

3.8 Examples from Everyday Life

Spirals with approximately the same thickness are created by rolling up and unrolling processes, such as the cross-section through a rolled up carpet or a Swiss roll.

The classic, of course, is the toilet paper roll. The outline is approximately a circle. Therefore, the toilet paper roll can be used to derive the formula for the area of a circle, given knowledge of the circumference formula ($U = 2r\pi$). The story goes like this: Many years ago, I attended a conference in P. A friend advised me to pack a roll of toilet paper. When I actually needed the paper, the roll in the backpack was compressed (Fig. 3.30).

Fig. 3.30

Toilet paper roll, original and compressed

Since the area formula for the circle, as the name suggests, provides an area, the approach $A = \alpha r^{2}$ is sensible. What is sought is $\alpha$.

The cross-section of the original roll was a circular ring with the outer radius $R$ and the inner radius $r$. For the circular ring area $A$ we get according to our approach:

$$ A = \alpha R^{2} - \alpha r^{2} = \alpha \left( {R + r} \right)\left( {R - r} \right) $$

(3.5)

In the compressed situation, there is a rectangle in the middle of length $\pi r$ (half the circumference of the inner cardboard cylinder) and width $2\left( {R - r} \right)$. At both ends, there is a semicircle with the radius $R - r$. The total area $A$ results from this:

$$ A = 2r\pi \left( {R - r} \right) + \alpha \left( {R - r} \right)^{2} $$

(3.6)

Comparison of Eqs. 3.5 and 3.6 yields:

$$ \begin{aligned} \alpha \left( {R + r} \right)\left( {R - r} \right) &= 2r\pi \left( {R - r} \right) + \alpha \left( {R - r} \right)^{2}\\ \alpha \left( {R + r} \right) &= 2r\pi + \alpha \left( {R - r} \right) \\ \alpha &= \pi\\ \end{aligned} $$

(3.7)

Thus, the area formula for the circle is obtained:

$$ A = \pi r^{2} $$

As further examples of approximately Archimedean spirals, Fig. 3.31 shows a damaged climbing rope and a tipped-over basket.

Fig. 3.31

Climbing rope. Toy basket

According to Chap. 2, nature primarily produces logarithmic spirals as a result of exponential growth. In contrast, Archimedean spirals are typically “man-made”, developed by people using various cultural techniques.

3.9 The Spirals of Pythagoras

The theorem of Pythagoras is usually illustrated with squares. The squares can be replaced by their inscribed circles, and the inscribed circles can be approximated by spirals (Fig. 3.32).

Fig. 3.32

Red = blue

Winding up and corresponding unwinding of the cathetus spirals illustrates the invariance of the area sum of the cathetus squares (Fig. 3.33).

Fig. 3.33

Winding up and unwinding

The area sum of the cathetus squares is thus invariant. In the limit case, however, one of the two cathetus squares is the same size as the hypotenuse square, and the other has shrunk to a point with an area content of zero. This also results in the equality of the area sum of the cathetus squares with the area content of the hypotenuse square in the general case (Fig. 3.34).

Fig. 3.34

General case (▶ https://doi.org/10.1007/000-br5)

3.10 The Beard of Archimedes

Archimedes used to annoy his scholarly visitors with the question of how large the red portion of the total circle area is in Fig. 3.35a [1]. The separating curve between the colors red and green is an Archimedean spiral.

Fig. 3.35

How large is the red area portion?

The areas can be resolved into concentric threads (Fig. 3.35b). The longest red thread is the one in the middle. It has the sector angle $\pi$ and in the unit circle the radius $0{.}5$. Its length is therefore $0{.}5\pi$.

Now one end of the red threads is attached to the horizontal black line and the other end is dropped (Fig. 3.36a). The red threads are thus combed vertically downwards.

Fig. 3.36

Archimedes’ beard

During combing, the area does not change. A red thread can be modeled as a circular ring sector and cut into an even number of equal parts. Reversing every second part eventually transforms the thread into a vertical line (Fig. 3.37).

Fig. 3.37

Combing

The figure resulting from the combing, the Archimedes’ beard, thus has the same area as the sought-after red area in the spiral.

The outline of the figure is a square parabola. In the coordinate system of Fig. 3.36b it has the equation:

$$ y = 2\pi x\left( {1 - x} \right),\;0 \le x \le 1 $$

(3.8)

For the sought area $A$ we calculate the integral:

$$ A = \left| {2\pi \int_{0}^{1} x \left( {1 - x} \right)dx} \right| = \frac{1}{3}\pi $$

(3.9)

This is one third of the circle area.

What proportion of the circle does the twisted moustache cover (Fig. 3.38a)?—The moustache is a twisted Archimedes’ beard (Fig. 3.38b). Its area is therefore also one third of the circle area. One can ponder the philosophical question of which side the middle and longest hair should be twisted to.

Fig. 3.38

How big is the moustache?