Quantized angles

The unit angle of the regular heptagon
In a regular heptagon all diagonals are drawn.
All sides and diagonals are extended in both directions.
Then arises the figure with a tangle of lines and angles:



This figure only contains angles 1, 2, 3, 4 or 5 × 180°/7.
I define the 180°/7 as the unit angle of the regular heptagon.
I display it as u₇.

All possible trangles in the regular heptagon are of the next shapes:
(1,1,5)u₇, (1,2,4)u₇, (1,3,3)u₇ and (2,2,3)u₇.
Between () you find the three angles of the triangle
expressed in the unit angle u₇.
You find the same numbers in the diagram.
The sum of the three angles in a triangle is 180° = 7 units always.
That is why in (for instance)(1,2,4)the sum of the three number is 7 again and again.
Indeed 180°/7 equals u₇. So 7u₇ = 180°.
The next diagram is a part of the previous figure.
In every angle you see the number of units u₇:



In nature and daily life regular heptagons are very rare.
Look at the 20 pence coin and the 20 euro cent coin: it is a regular heptagon.
The edges has 7 notches in favor of blind.

Unit angle of any regular polygon
It is a beautiful fact, that the mentioned
quality is true for every regular polygon.
Each regular polygon has its own unit angle.
It only depends om the number of angles. Being n the unit angle equals 180°/n.

Summary
As in a regular polygon all diagonals have been drawn,
each angle is a multiple of 180°/n.
Physicits would say: the angles have been quantisized.

I was very surprised that this property had never
been discovered in the past two millennia!

Some already known properties
The sum of the angles of a regular polygon equals (n - 2)180°.
So each angle of an empty regular polygon equals (n - 2)180°/n.

The number of diagonals is n(nn- 3)/2.

The next table shows some examples:

diagram	name	number of angles and sides	number of diagonals	sum of the angles	size of the angles in °	unit angle in °	related diagram
	equal side triangle = regular triangle	3	0	180	60	60	delta
	squair = regular quadrangle	4	2	360	90	45
	regular pentagon = pentagon	5	5	540	108	36	pentagram or 5-pointed star only diagonals
	regular hexagon	6	9	720	120	30	Star of David Christmas star
	regular heptagon	7	14	900	128 + 4/7	25 + 5/7	€ 0,20 coin
	regular dodecagon	12	54	1800	150	15	(analog) clock
	regular 360-gon	360	64260	64440	179	1/2	almost circle
	regular n-gon	n	n(n-3)/2	180(n-2)	180(n-2)/n	180/n

the general proof has been based on the next geometrical properties.

• 180° = n × u_n
• angle + supplementary angle = n × u_n.
• opposite angles are equal in measure.
• the measure of the central angle is exactly twice the measure of the inscribed angle.
• the inscribed angle = the unit angle
• the sum of the angles in a triangle is 180° = n × u_n
• the sum of the angles in a quadrangle is 360°
• the sum of the angles in an n-gon is (n - 2) u_n
• the regular polygon mirror symmetrical and rotational symmetrical
• that is all folks!

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