Zef Damen Constructions with ruler and compass

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 My crop circle reconstructions are "ruler-and-compass" constructions, as far as possible. What are ruler-and-compass constructions? Ruler-and-compass constructions only use undivided rulers, and compasses. The ruler can only be used to draw lines between (or passing through) two existing points. Compasses can be used for drawing circles or arcs from an existing center point and another point determining the radius. Besides that, compasses can also be used for transferring the distance between two existing points. It can be transferred to a third point to create a new point at an equal distance, or it can be used to draw a circle or arc with the distance as the radius. (Note: P.E.Zimourtopoulos of the Democritus University of Thrace, Greece, reminded me, that strict ruler-and-compass constructions cannot use compasses in this last mentioned way. Whenever a compass is not in the act of drawing a circle, it "evaporates", it disappears into nothing! In "Copy a Circle", however, I show that it is possible to copy a given circle to an arbitrary given point with the only use of such an "evaporating" compass – and drawing lines between two points –, thus according to the strict ruler-and-compass method.). Every construction starts with two arbitrary points, used to draw a line or a circle (or both). Then every extension (next steps of the construction) uses points already given or created by previous construction steps. New points come into existence every time lines and circles intersect. Not all patterns can be constructed by (strict) ruler-and-compass rules. For instance, a regular heptagon – 7-sided polygon – can not be constructed in this way. From a mathematical law, it has been known, that a regular n-sided polygon can be constructed by ruler-and-compass, if and only if n is a finite product of different numbers from the set of powers of 2 (2, 4, 8, 16, ...) and primes of Fermat (3, 5, 17, 257, 65537, ...). According to this law, regular n-sided polygons are constructible for n = 3 (equilateral triangle), 4 (square), 5 (pentagon), 6 (hexagon), 8 (octagon), 10 (decagon), 12 (dodecagon), 15, 16, 17, 20, ...; however, n-sided polygons with n not present in this series, are not constructible (n = 7 (heptagon), 9 (nonagon), 11 (undecagon), 13, 14, 18, 19, ...). In the next sections, a number of constructions will be shown. Within these drawings, the following symbols have been used: Center point of compass for construction of circle or arc. Direction for construction line, circle or arc. Constructed point. Arbitrary or special point. In all drawings, elements will be numbered by the step in which they are introduced. How to bisect a given angle? How to divide it into two equal parts? How to construct a line perpendicular to a given line, and dividing it into two equal parts? How to construct a line perpendicular to a given line, passing through a given point on this line? How to construct a line perpendicular to a given line, passing through a given point not on this line? How to construct the horizontal and vertical centerlines of a given circle? How to construct an equilateral triangle, given one side? How to construct an equilateral triangle, inscribed in a given circle? How to construct a pentagon, a regular 5-sided polygon, inscribed in a given circle? How to construct a hexagon, a regular 6-sided polygon, given one side? How to construct a hexagon, a regular 6-sided polygon, inscribed in a given circle? How to construct a heptagon, a regular 7-sided polygon? How to construct an arbitrary regular N-sided polygon, inscribed in a given circle? How to construct any circumscribed regular polygon, given the corresponding inscribed regular polygon? How to construct a line parallel to a given line, passing through a given point? How to construct a line parallel to a given line at a given distance? How to divide a given line into an arbitrary number of equal parts? How to construct a circle tangent to a given circle at the near side, from a center inside that circle? How to construct a circle tangent to a given circle at the far side, from a center inside or on that circle? How to construct a circle tangent to a given circle at the near side, from a center outside that circle? How to construct a circle tangent to a given circle at the far side, from a center outside that circle? How to construct a circle tangent to a given line? How to construct a line tangent to a given circle? How to construct a line tangent to two given circles of different sizes? How to construct a line tangent to two given circles of equal sizes? How to construct a circle passing through three given points (not on one line)? How to construct a "two-points" circle, passing through the two end-points of a given centerline? How to construct a Vesica Pisces (two equal circles passing through each other's center), given one circle? How to construct a Vesica Pisces (two equal circles passing through each other's center), with a given radius and a given common center? How to construct a Vesica Pisces (two equal circles passing through each other's center), inscribed in a given circle? Copyright © 2001-2017, Zef Damen, The Netherlands  Nederlandse versie Last updated: 3-August-2017 Since 1-February-2005