free hit counters

Heptagon

HEPTAGON

Modes of Inside Constructions

The HEPTAGON – not to speak of the number seven and its importance and values, which was since ancient times always the subject of many discussions. It will suffice to mention that it was considered impossible to draw only with a compass and unmarked straightedge or depicted in any other way because of being seen as an infinite numeral of degrees – and then again we come across the misunderstandings concerning decimal number 360 ÷ 7 = 51,428571… that is to say, we don’t ariticulate it in parts (fractions) since 51 is an integer and 0,428571 = 1:7 = 0,1428571 namely 1/7 x 3 = 0,1428571 x 3 = 0,428571 = 3/7 therefore 360°/7 is read as: 51 degrees and three sevenths (three sevenths of a degree = integer 51 and 3/7 ° = 51 3/7° – angle magnitude, which after ascertaining again means: 7 x 51 = 357 + 7 x 3/7 = 21/7 = integer 3 = 357 + 3 = 360°
(this is simply a good-to-.know explanation to bear in mind when operating in a divisible/fractional mode).

However we will focus on the various possibilities, the modes of heptagon construction (truncated due to limited space), using the hitherto described knowledge in our previous chapters and supplements for kids, only applying inside divisions of the initial (basic) circle of the hexagon and putting aside products that arise outside of the circle. Thus, a series of modes of constructing the heptagon and dividing its arc of seven parts and onward to 14, 28, 35, 42 parts to the conclusive part of the square of seven (72) 7 x 7 = 49 parts to the next progression of seven; and of course by applying only the compass and unmarked straightedge.

* * *

First mode: We have already learned it in the chapter for kids. The circle, divided by radius into six parts, one equilateral triangle the magnitude of its sides the same as the circle’s radius – its height as the radius …

* * *

… divides the arc of the circle into seven equal parts (from one pole) – the heptagon.

* * *

Second mode: (we got acquainted with it in the chapter for kids) – the radius divides the circle into six parts – by inscribing lengths of radius from pole to pole – a hexagon – inscribe circle to the hexagon …

* * *

… and that is a rule. The radius of the inscribed circle of the hexagon divides circle into seven equal parts – the heptagon

* * *

Third mode: With radius (randomly taken) we describe a circle. We divide it with radius and inscribe it (from pole to subtended pole). The circle is divided into six parts.

* * *

Inscribe a six-sided star-shaped polygon (every other pole)

* * *

The sides of the six-sided star-shaped polygon are already divided in two by diameters. Enter into compass one half of the length of a side of the six-sided star-shaped polygon.

* * *

That value is the radius that divides the described (basic) circle into seven equal parts – the heptagon.

* * *

Fourth mode: The circle and its star-shaped polygon. Using the intersection points of the star polygon (left and right of the center) and the center itself, we bisect it. Actually, including the vertices (lower and upper) we divided the circle into four sections.

* * *

… From the four points we describe semi-circles.

* * *

We connect the intersecting points of these lines of symmetry with rays (subtended). We just divided the circle into six parts (the star polygon) and eight parts (2 x 4) where we only displayed the second quadrangle bisecting 6 and 2×4. We circumscribe the circle.

* * *

That radius, made /geometrically) by additions, divides the basic circle into seven equal parts. A heptagon.

* * *

Fifth mode: The ancient geometrical (truncated) mode of the hexagon with diameters and its six-sided star-shaped polygon.

* * *

Hexagon within the star polygon. We inscribe its depicted circle.

* * *

Divide inscribed circle of the inside star-shaped hexagon with divisible circles of its radius into six parts, as well as the divisible circles.

* * *

Radius of line segment of basic circle – Divisible circles of hexagon inside of the star polygon – Their exterior points of intersection – That radius divides the inscribed (basic) circle into seven equal parts – Heptagon.

* * *

Now divide with same radius (radius of heptagon) from the two subtended poles (2×7 = 14) into 14 parts of the basic circle (its arc).

* * *

From the three poles (every other one) of the six-sided polygon (3×7 = 21) we acquire 21 equal parts.

* * *

Using the intersection points of the star polygon, from four poles (4×7 = 28) we acquire twenty-eight parts of a circle.

* * *

From all six poles of the hexagon (6×7 = 42) we divided the arc of the circle into 42 equal parts.

* * *

Let’s recall the construction of the pentagon … The circle – Its star-shaped polygon – Division into twelve – From peak pole radius of the peak pole – Every other one of the hexagon (every fourth one of the dodecagon).

* * *

Hence, from the peak poles of these radii we divide the circle from each of the 12 poles. Then we take the radius of the first exterior intersection points and with it we divide the basic circle into 5 equal parts. The pentagon.

* * *

Now we inscribe the heptagon radius.

* * *

With a heptagon radius, from the pentagon poles of the basic circle, we perform the division of the basic hexagon circle from its peak pole (5×7 = 35). We divided the basic circle into 35 parts. This progression can be continued, 2x5x7 = 70, and so on. …

* * *

Quadratic mode: We will apply the fifth ancient geometrical mode to construct the heptagon (any other mode can also be applied – the essential factor is the circle of the heptagon).

* * *

When we acquire this circle, we inscribe its star-shaped six-sided polygon.

* * *

It also has a hexagon within its star-shaped polygon. We use the inscribed circle to acquire its heptagonal radius, because…

* * *

… its radius divides the (described) basic circle of the hexagon into 49 parts (7×7 = 72). We only used the peak pole, thus we could continue the (quadratic) progressions.

* * * *

AFTERWORD

We will stop here even though we have not referred to the exterior radii of the heptagon that form progressions of dissimilar star-shaped seven-sided polygons and we did not draw them in full circles (due to the limited space of the A4 format) as sacred geometry basically requires. Thereby we have surely omitted some serials of essential information, but this is also an essential part of the basics – an introduction to hundreds and hundreds of years of untouched domains of ancient sacred geometry. Sure enough, in these presentations I left out the so-called “pyramidal code”, but for the time being this is not our theme. It could even provoke certain turbulent reactions from persons who to this very day call the largest pyramids “pyramids of the sun”, forgetting that ancient peoples established their beliefs “from new moon to new moon”, and that their incidence was not 360°÷7 but 52°. Why? And why without a peak? But, this will be dealt with on some other occasion.

Prior to that, new cognitions are of greater importance: squaring the circle, for example, as in ancient geometry (for adults) and in a simpler (shortened mode) for kids. Then coming up are new chapters: reckoning the height of equilateral triangles (with compass only) and the ancient enigma of doubling the cube (with compass and unmarked straightedge).

2 Responses to “Heptagon”

  1. vishal says:

    thank you 🙂

  2. Elaine Stewart says:

    What is 70 times 7 mean.it’s all over myths legends bible etc.Thank you

Leave a Reply

Powered by WordPress | Designed by: suv | Thanks to trucks, infiniti suv and toyota suv