A better angle unit
We have already the angle unit degree. It is defined by:
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.
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).
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:
Thus, we can have the angle units: turn, radian and the unit 1 turn/240.
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 ¯) 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.
See also the other multiple values of turn/8:
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,
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:
- symbol: = (U+003D EQUALS SIGN)
- symbol: ≡ (U+2261 IDENTICAL TO, ≡)
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.
- - 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.
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:
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.rad = ", Turn.rad, " rad"
echo "Turn.deg = ", Turn.deg, " deg"
var angle: float = 120
echo "120 deg = ", toRad(angle), " rad"
echo "Turn.rad / 4 has: ", toDeg(Turn.rad / 4), " deg"
echo "Right angle: sin(toRad(60)) = ", sin(toRad(60))
Save it as test.nim file and compile it.
If you run it, you will see in a terminal:
Turn.rad = 6.283185307179586 rad
Turn.deg = 240.0 deg
120 deg = 3.141592653589793 rad
Turn.rad / 4 has: 60.0 deg
Right angle: sin(toRad(60)) = 1.0
This is my point of view: we need the elegant solutions! And the unit turn/240 is an "elegant" solution. However, it is a better angle unit!
Copyright©2019 Lulzim Gjyrgjialli. A better angle unit.
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