And in his hand
He took the golden Compasses, prepar’d
In Gods Eternal store, to circumscribe
This Universe, and all created things:
One foot he center’d, and the other turn’d
Round through the vast profunditie obscure,
And said, thus farr extend, thus farr thy bounds,
This be thy just Circumference, O World.
Thus God the Heav’n created, thus the Earth.
Paradise Lost VII.225
Practical geometry, the ancient, godlike skill of creating an orderly, rational world on paper, had a rough time of it in the 20th century. Mass produced consumer goods and ever increasing computing power brought hand drafting to the brink of extinction. Hand tool woodworkers researching the history of their craft have created a limited revival of traditional layout methods, but good luck getting any of them to help you draw a Sonnenrad. No doubt a computer generated version would be easy enough to produce, but honestly, everyone spends too much time staring at screens these days. Drawing by hand is a high agency activity, a real skill that once mastered can be replicated in any location on any medium in any size. The bobbles inherent in handcrafted work are also reminders of human frailty and imperfection, the true divide between the digital and material worlds. A certain type of autiste will repeat this exercise over and over seeking perfection, so here are a few tips to expedite the process.
The Sonnenrad can be constructed with a ruler, large compass, dividers, and a protractor. Fixing your paper in place on a work surface with tape makes precise measurements easier. A high quality or vintage German or Japanese compass is less frustrating to work with because it will hold its settings much better than modern plastic junk, and a set of dividers with two metal points is easier for stepping off segments than a regular compass. The lighter your layout lines, the easier it will be to erase the unnecessary portions in the final step. If you have not had experience drafting by hand (this ain't AutoCAD), skim a geometry textbook and do some exercises from a vintage layout book first. Useful basic operations include bisecting a line segment, bisecting an angle, trisecting a 90 degree angle, and raising a perpendicular.
These plans are based on the proportions of the Sonnenrad components instead of specific measurements, so they can be expanded to any size. The main directions give the simplest way to perform each operation, while additional but superfluous details can be found in the technical notes at the end. There is nothing complex about the Sonnenrad; success is mainly a function of precision, a steady hand, and an orderly accounting for the various points and angles involved. Do practice on paper before you invest in a hoplite shield. That said, if /pol/ got enough Anons together with lawnmowers, tape measures, and marking paint they could cut Sonnenrad crop circles into a wheat field.
SONNENRAD LAYOUT PLANS
For a practice Sonnenrad, a 2.5 inch diameter center circle will fit nicely on a 8.5x11 sheet of printer paper. In the directions I have labeled all of the points used for making rays. If you have a good eye and you grasp the concept of what points need to be connected, there is no need to label everything. If you make a template to pass on to others it is good geometric practice to label all points as a courtesy.
Parts of the Sonnenrad:
The center circle (1)
The inner circle (2)
The partial circle (3)
The outer circle (4)
The inner rays (5a)
The outer rays (5b)
Draw a straight reference line across the length of your page passing through the center point of the paper. (11 inch long line 4.25 inches up from the bottom of the page if using standard American printer paper)
Set your large compass to the desired radius of your center circle. (1.25 inches)
Place your compass pin at the center point of the page on your reference line and carefully inscribe the center circle. Leave the compass set to this radius.
Label the point where your compass pin touched as point A. This is the center of the entire Sonnenrad.
Label the 3 o'clock point where your reference line and circle perimeter meet as point B.
Using the large compass with the same radius again, set the pin at point B and mark a dash on the reference line. Label as point C.
Set the pin of the large compass at point C and mark a dash on the reference line. Label as point D.
Reset the large compass to the length AD and inscribe another circle with center A. This is the outer edge of the Sonnenrad. Thus the full diameter of the Sonnenrad is three times the diameter of the center circle.
Divide the line segment BD into eight equal parts, placing dash marks on the reference line. This can be done with a ruler or by using a compass to bisect segments BC and CD and then again bisecting the resulting segments.
Label the dash marks 0-8 from B to D, so that B= 0, C=4, and D=8.
Set the large compass with the radius A1 and inscribe a circle with center A.
Reset the large compass with the radius A2 and inscribe a circle with center A. These two lines define the edges of the inner circle of the Sonnenrad.
Reset the large compass with the radius A4 and lightly inscribe a circle with center A.
Reset the large compass with the radius A5 and lightly inscribe a circle with center A. These two lines define the edges of the partial circle of the Sonnenrad and will be partly erased upon completion.
Reset the large compass with the radius A7 and inscribe a circle with center A. This line defines the inner edge of the outer circle and completes the circular parts of the Sonnenrad.
The perimeter of the center circle must be divided into 24 equal parts. Take great care in these steps because they determine how wide each of the sun rays will be. The more precision in dividing the circle exactly, the more uniform the final rays will be.
Measure off two 15 degree angles with a protractor starting from line segment AB and moving counter clockwise.1
Starting at B and moving counter clockwise, label the two points E and F on the perimeter of the center circle such that the angle BAE= 15 degrees, EAF = 15 degrees, and BAF = 30 degrees.
Set the dividers to the distance EF and step off this distance all the way around the center circle. There should be 24 equal segments. Leave the dividers set to this length.
Draw reference lines inside the center circle by lining up point A with point E, point A with point F, etc. Continue labeling the next ten points counter clockwise around the circle in order: G, H, I, J, K, L, M, N, O, and P.
Locate the opposite angle of EAF and label E prime and F prime such that angle EAEprime equals 150 degrees, not 180 degrees.
Line up points E and E prime with a straightedge and draw a line from the perimeter of the center circle out to the outer edge of the partial circle. Do the same with the points F and F prime.2
Locate the opposite angles of GAH, IAJ, KAL, MAN, and OAP, label their prime points just like EAF, and draw the rest of the rays by lining up each point with its prime from its opposite angle and extending the lines out to the partial circle.
The ray lines are straight, so that the spaces between the center circle, the inner circle, and any two rays form mixtilinear trapezoids. The illusion of perspective makes the rays seem like they are narrowing, but if you check them with the dividers they are actually the same width at each end.
To space the outer rays, the perimeter of circle A4 must be divided into 48 equal parts, each part the width of a ray. Circle A4 is twice the diameter, and therefore twice the circumference, of the center circle. The rays extended out already account for 12 of the 48 parts, so the space between each pair of rays must be divided into three equal parts.
Use the dividers set to the length EF and step off three spaces between each pair of inner rays on circle A4.
Label each point, 1-48, on the partial circle starting with the one formed by line segment AE and moving counter clockwise.
Each outer ray is equidistant from the inner rays on either side of it. Thus, every fourth section on the partial circle gets an outer ray. In our diagram this includes sections 3-4, 7-8, 11-12, 15-16, 19-20, 23-24, 27-28, 31-32, 35-36, 39-40, 43-44, and 47-48.
With the straightedge, line up points 3 & 28 and draw a line from the partial circle to the outer circle. Do the same for points 4 & 27, forming two short rays on opposite sides of the Sonnenrad.
Repeat with points 7 & 32, 8 & 31; 11 & 36, 12 & 35; 15 & 40, 16 & 39; 19 & 44, 20 & 43, 23 & 48, 24 & 47.
All of the elements of the Sonnenrad are now in place. All that is left is to fill in the circles and rays, erase all labels, and erase the short line segments of the partial circle between the rays.
Oh, and don't forget to play some Vivaldi while you work on this.
The length EF is the width of the sun rays and is also approximately the width of the inner circle, the partial circle and the outer circle of the Sonnenrad. Because the diameter of the center circle is equal to the length BD, 1/24th of the circumference of the center circle is approximately equal to 1/8th of BD [diameter x pi /24 ≈ BD/8]. The length EF that is used to set the width of the rays is technically not 1/24th of the circumference of the center circle, because a compass set to the length EF actually measures the chord EF, not the arc EF. The chord EF is the length of one side of a tetracosagon (regular 24 sided polygon) inscribed within the center circle. Side length of a regular polygon = 2r sin 180/n where r is the radius of the center circle and n is the number of sides of the inscribed regular polygon. For the tetracosagon this simplifies to .261052384r. ↩
Each ray is defined by a pair of parallel lines that pass on either side of the center point A. That is, each ray edge line is a chord of the outer circle. Since the width of a ray is equal to the distance EF, each line defining the edge of a ray is .5EF away from point A. Put another way, each ray edge line is a tangent to a circle of diameter EF with center A. ↩