# A better angle unit

We have already the angle unit degree. It is defined by:

1° = turn/360

Is the definition of the unit degree the best definition we can have?

An angle represents a rotation. It represents also a physical dimension. A physical unit must be easy to use. An angle is also a geometrical quantity.

The angle unit must be easy to construct!

Can you construct an angle with the value 1° only by a compass and a straightedge? Nobody learns how to do it! There is always only the definition of the angle unit degree. But I know how to construct an angle with the value 1.5° only by a compass and a straightedge! If the angle 1.5° is easy to construct, it must be the angle unit. See first the angle values in the fig. 1 below. Fig. 1 – Angles represented by the new angle unit turn/240.

You can see that the right angle (90°) has 60 new angle units. All other angle values in the fig. 1 are easy to understand.

All my images have the accurate angle unit turn/240 !

The angle unit degree defined by turn/360 is a very small unit. The angle unit turn/240 is also a very small unit, but it has a better size for a drawing. Any angle is as you know radius-invariant. If the circle radius is bigger, the arc of the angle unit is bigger. In this case we have a better visibility of the angle unit. You can zoom an image to see the difference. The construction of the angle unit turn/240 is given by the fig. 2 below.

The angle unit degree is defined by turn/360. Thus, you can forget this image! We can define a new angle unit (equal to 1.5°) directly by a turn.

1 angle unit = 1.5° = turn/240

I have to use the unit degree for my explanations because the new angle unit turn/240 needs an acceptance. You can get the angle 36° = turn/10 by the golden ratio length 1/φ (the length BI = BJ in the image above), where φ is the golden ratio (approx. 1.618). The angle 30° = turn/12 is easy to construct. The difference angle 36° - 30° = 6° is also not a problem. Thus, the angle 6°/4 = 1.5° is easy to draw. Every subunit of a turn can be defined directly by a turn!

A turn is the natural reference angle.

See the values below:

1 turn/10 = 36°

1 turn/12 = 30°

1 turn/60 = 6°

1 turn/240 = (1/4) · (1 turn/60)

1 turn/240 = 1.5°

A turn is also the circle constant (see: Circle constant is a turn). Thus, we can define the angle unit by a circle and a turn (1 turn = C/r). Fig. 3 – Angle unit definition: 1 angle unit = turn/240.

Note:
This image (angle_unit_def.png) has the dimensions:
712px × 440px because 712px / 440px = (8 · 89px) / (8 · 55px) ≈ 1.618.
You can use the multiple values of the dimensions 89px and 55px to get approx. the golden ratio of an image:
(n · 89px) / (n · 55px) = 1.618...

This is like a classical construction! I have not a name for this new angle unit (1 turn/240)! But it has a reasonable definition! Keep it simple! You can see that the angle unit definition 1° = 1 turn/360 is not the best definition. We can have a better angle unit definition. See the values of the angular unit 1 turn/240:

The angle unit turn/240 is easy to use. This angle unit can be constructed also in the second way: Fig. 5 – Construction of the angle unit turn/240 by the circle constant and the golden ratio (angle_unit_constr.png).

Thus, we can have the angle units: turn, radian and the unit 1 turn/240. Fig. 6 – Angle unit and the circle constant 1 turn = C/r.

Yes! Circle constant is a turn in a circle!

See a protractor with the angle unit turn/240 :

You can see less angle ticks on this protractor than on a protractor with our old angle degrees. Temporarily I am using the symbol overline ¯ (Unicode: U+00AF, or the decimal value &#175;) for this new unit. The symbol of a measurement unit is very important. In this case, we need a new symbol! However, the new angle unit turn/240 can be called also "degree" (maybe with a notice that it is a "new degree") ! But, the others have to say also their opinion. Let us have the angles:

1·turn/8 + 2·turn/8 = 3·turn/8

If the angle unit degree is 1° = 1 turn/360, we get:

45° + 90° = 135°

But, if the angular unit is 1 turn/240, we get:

30 + 60 = 90 angle units.

Simple sometimes means beautiful. Bisection of an angle is a division of the angle into two equal angles. As you know, it is not difficult to do it by a compass and a straightedge (ruler). Thus, let us see the half of the angle turn/8. If we use the angle unit degree, it is:

turn/16 = 360°/16 = 22.5°

We get a better result by: turn = 240 units

turn/16 = (240 units)/16 = 15 units

We get still an integer value. The number 360 has more divisors. But, who needs the divisor 9 or 18 of the number 360? There is no need to talk about the other divisors of the numbers 240 and 360! The number 360 is also a bigger number! If the world is divided into 24 time zones, every time zone can have 10 angle units turn/240 of longitude (much better than 15°). Do not forget that the unit turn/240 is also easy to construct only by a compass and a straightedge! The angle unit turn/240 is a better angle unit than our old angle unit degree defined by 1° = turn/360 !

If you say "angle unit definition", you have to say also the word "turn". A turn is always our natural reference angle.
We have to emphasize that the turn (a full rotation) is a "natural angle unit".

If you see turn = 240 unit, it is not the definition of turn. It is only a mathematical equality (needed for a calculation)! Also turn = 360° is not the definition of turn!

For the definition of a turn (full rotation) there is no need even for mathematics or any number! A turn is a physical constant, and it is very simple to define it. A turn is simply a full rotation around a position until you point in the same direction again.

A definition can give us a (new) meaning of something. And every equality is not a definition. Equalities are used for calculations. This is the reason why we have in this case two different symbols:

equality
symbol: = (U+003D EQUALS SIGN)

definition
symbol: ≡ (U+2261 IDENTICAL TO, &#8801;)

You may not define a turn by a subunit of a turn!
You may define an angle unit by a turn:
1 unit ≡ turn/240 (or 1° ≡ turn/360)

If we have the definition 1 angle unit ≡ turn/240, it is turn = 240 units.

The angle unit degree is also an angle, and it is a subunit of a turn. The first thing what a child learns is the rotation and the full rotation! In one direction is a parent, and in another direction is not! We already know what is a turn. Thus, the equality turn = 360° is not a definition! But, we have the definition of the unit degree: 1° ≡ turn/360.

We can define something new by something what we already know!

Forget the calculation number pi and the fraction pi/180. The calculation 1° = pi/180 is wholly unacceptable! The fraction pi/180 is only a calculation number and nothing else! It is not the definition of the angle unit degree!

An angle unit must be by definition an angle!

If the angle unit is a subunit of a turn, it is an angle. It represents also a rotation.

degree
- The basis for the definition of the angle unit degree is the constant angle turn. The unit degree is a subunit of a turn. It is good to give a direction, and it is good for orientation.

- The basis for the definition of the angle unit radian is the fact that an angle is radius⁠-⁠invariant. The unit radian is an angle unit, but it is not (per definition) a subunit of a turn. It is good for mathematical calculations of the arc lengths.
If "s" is the arc length of a circle with a radius "r", the angle defined by 𝜃 (theta): 𝜃 = s/r is radius⁠-⁠invariant. If s = r, it is 𝜃 = 1 radian. In this case, a turn has approx. 6.28 rad. We have two different angle units degree and radian for two different views of the angle. But the turn is the natural unit!

Angle units:  deg  (min=-240, max=240)

Angle: 0
Angle: 0

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### Programming turn

And here is something for the programmers. The Nim programming language is a new language for me. I have just started to test it (Nim Compiler Version 1.0.4, Compiled at 2019-11-27). It is easy to understand this language. The Nim code is like a pseudocode of a tutorial! But you can compile it! And the Nim programming language has its modules! See the code below:

File: turn.nim

I like the const section above (it declares the constants). The parts with "*" (U+002A ASTERISK, star) are visible outside of this module (file). If we decide to name the unit turn/240 by another name, we can make the procedure toDeg() obsolete, and we can define a new procedure for a new unit name (for example: toAngleUnit() or something else). Of course, in this case, also deg in Turn.deg must be changed. The implementation code (without a star symbol) in the module turn is not visible outside of this module. We can change it without a notice! You can save this code as a turn.nim file. This code can be tested by the code below:

# ------------------
# File: test.nim
# ------------------

import math # We need sin()
import turn # Turn, toDeg(), toRad()

echo "Turn.deg = ", Turn.deg, " deg"

var angle: float = 120

Save it as test.nim file and compile it.   See: Nim
If you run it, you will see in a terminal:

Turn.deg = 240.0 deg
Turn.rad / 4 has: 60.0 deg 