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The Cube part 1

The CUBE
– Ancient Geometric Partitioning –

CUBE (PART 1)
– ancient geometric partitioning –

Constructing in ancient geometry (only with a compass and unmarked straightedge) opens the door to a vast area with a sequence of products, so that it cannot be incorporated on just a couple of pages of these presentations. With respect to the preciseness of straight line (shortened) drawing, it is far more precise, so that this small-scale but targeted amount that we will present is beyond doubt essential for the columns of measurements and in some examples for enigmas and their solutions as well as for new and fresh queries. Presernt-day geometry pays no attention to them (examples that we will soon present: the pyramidal code, why are some pyramids without vertices, the sanctity of number 108, progressions of the products of measurements or the so-called “eye” that is often encountered in Egyptian culture, however in a stylized form.
However we will start with the two basic columns: 2 and 3, through the cube or hexahedron or the circle divided into 6 parts by its radius – the hexagon – the basis of ancient geometry whose circle constitutes the base as well as its hexagon, radii and diameters and the diagonals of its cube and its hexagonal star-shaped polygon (the Star of David). Everything arises from these basic elements, and we must always bear that in mind and before our eyes – the harmony that is embedded in the entirety of natural science as its omnipresent elementary nucleus. Although such observation adverts to a broader approach to antiquity than ancientness itself, referring to other areas (natural and unnatural) apart from the one “assigned” to me, namely the pure and simple path of geometry that ought to clarify what is unclear, make possible what is considered impossible, correct the incorrect and (I repeat) do so only with a compass and unmarked straightedge, or, as I have the habit of saying, “only with a cord and straight rod”. Hence, using historical data and relegated to the Invisible one to lead me, “to incite me or push me” when I face difficulty on this lonely path as is recorded somewhere in the ancient Egyptian inscription „…ah, how lonely the road between the stars”, which might be the origin of RAF’s slogan “Through struggle to the stars”.
But, let’s move on to present numbers 2 and 3 from the aspect of ancient geometry.

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Thus, the circle divided by its radius into six parts – a hexagon – a cube and its star-shaped hexagonal polygon – a hexagon and its inscribed circle (the magnitude of diameter: two quarters of the radius).

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We divide the inscribed circle with same divisible radius into six parts.

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And the intersection points of these circles divide the sides of the hexagon of the basic circle in two parts. We connect these intersections with line segments – vertically – subtended – left – right. We have divided the hexahedron into the cube of (2x2x2) – into eight parts.

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We continue with the division of the divisible circle – six parts and once again we use the same radius as the inscribed circle of the star-shaped polygon and thus acquire a division outside of the circle.

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We connect the subtended intersections with dotted lines.

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The dotted lines intersect the sides of the cube that is divided 2x2x2 and connect these intersections. We have now acquired a cube of 4x4x4 or 64 parts of the cube. We will display an example of this.

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The start of the “eye”. Being the column of number 2, the diameter of its star-shaped polygon is divided into quarters. Then we take into our example ¾ of the subtended poles of the basic circle.

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We acquire a new radius that divides the basic circle of the hexagon into 21 parts (from one pole alone) and we may surmise what we could get with that radius if we went on with its division from 2, 5, 7, 8. However, let’s make an introductory pause on this example.

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Let us now move into the sphere of the second column of measurement: number 3. Its base –the circle and its hexagon – the cube.

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Circle’s hexagonal star-shaped polygon.

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The described circle of the hexagonal star-shaped polygon.

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Divided by its divisible circles.

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Let’s also divide them with the same radius.

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We have terminated their division. We could divide the sides of the cube of 2x2x2 to equate the cube of 2 by connecting the subtended divisible vertices through the center of the circle.

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But we have decided to apply another way (vertical – oblique – oblique) – vertical first of all.

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Thereafter oblique (along the sides of the hexagon).

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Thereafter oblique to subtended side.

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We connect the intersections on the sides with line segments in the same way – the cube is divided 3x3x3 = cube of 3, and the diameter into sixths (6 parts). We could go further on with 2, 3, 5 or 7, but before that one more example.

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If we continue the division of the circle (only from the vertices – poles of the basic circle) we acquire the intersections – external to the circle of the first row of the 6 divisible ones.

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This is a new – bigger radius. Start with division from the peak pole of the basic circle (from the point of division on it to the next point) until we finally return to the starting point – the peak pole of the basic circle.

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We acquired a product of ancient geometry’s manner of division – a radius that divides the basic circle into 36 parts or 10° (which is just one of the products of the series).

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So we carry on with the division of these divisible circles of the described hexagon of the basic circle.

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We will apply pattern:  vertical – oblique – oblique.

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Now we have correctly divided the cube along with a series of control points of the cube of 3, and the diameter into sixths (regardless of the fact that we are leaving the fourths of the arcs of 6 semicircles for the next sequel), but we will do so when we deal with the “pyramidal code” of pyramids without peaks and their geometrical implication. So much at present concerning the introduction of Part One of the Cube.

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