Compass equivalence theorem

The collapsing circle or Euclidean circle is a mathematical consideration, which describes a circle that snaps shut when lifting from the leaf.

Euclid used in his geometry collapsing compass. In Proposition 2 of Book I of the elements, he shows how you can still recover transmitted any distance with such a compass and a ruler, so the equivalence of the construction with ruler and collapsing and non- collapsing circles.

Problem and disambiguation

In constructions with ruler and compass usually a non- collapsing compass is used. In the original design problems of Euclid, it was considered by a collapsing circle whose radius from the leaf collapses when lifted, ie, can not be held to draw another circle with this radius.

It should be noted that a collapsing circle is not really a circle, but rather a mathematical consideration. According to Euclid exist two points A and B is always a ( constructible with a ruler ) straight line passing through the two points, and two circles, one to A and one to B, with the radius of the path from A to B.

Thus, one thus obtains almost all kinds of constructions, which you can do with a pair of compasses and a ruler, except for the tapping a line AB with a compass and drawing a circle with this radius around a third point C.

So, one can imagine a collapsing circle as a circle, which you put the needle on a point, then put the pen tip on another point and can draw a circle. However, after drawing the circle breaks the radius found together, so the circle snaps together, so that one set at this radius not put him on another point and can draw a circle the same Radiusses - unless there already exists a point from the new point already has this radius.

In fact, however, the result is that by further steps with a collapsing compass and a ruler also such circuits can be constructed which can be drawn with the radius as the distance between two points to a third point. Proof see below.

Mathematical explanation for constructions with ruler and compass

Is a set of points in the set of all lines in which pass through at least two points from the set of circles, the centers of the points and whose radii are equal to the distances from two points.

Then the set of all points that can be constructed by ruler and compass constructions is using the following operations:

For it makes no difference whether you're working with a collapsing or non - collapsing circle, because all points from which can be constructed using a non- collapsing circle, can also be constructed with a collapsing compass. Proof see construction below.

However: Now let the set of all points of M by collapsing constructions with a straightedge and compass and the set of all points that can be constructed from M by structures with a non- collapsing compass and ruler in one step.

Then: because even for a three-element set of points A, B, C is the intersection of a straight line through A and B with a circle with radius C, although in but not in. The construction of this circle to C requires a collapsing circle a few extra steps ( see below).

Constructing a circle around a point with distance between two other points as radius with a collapsing compass

In order to show that a collapsing compass and a ruler the same points can be constructed as a non - collapsing compass and a ruler, it suffices to show that with kollabierendem ruler and compass construction of a circle around a point with distance between two other points is possible as the radius. Cuts from such a circle with a line or with another (possibly just such ) circuit are then readily possible because already two circles (or circle and the straight line ) can be constructed, so accordingly their cuts.

Evidence

Given three points A, B, C are the line through A and B and A and C already drawn in the drawing. The line through B and C is not needed.

The objective is to construct a circuit to C, as the radius of the track length, in order to construct a point of intersection with the straight line. For two parallels have formed will, one each for each of the two existing lines. The first parallel ( to ) is intended to pass through the point C, and the second parallel ( to ) to pass through the point B.

For the construction of the first of these parallels, a circle is drawn with radius around the point C. This meets the line in A and another point D.

( If is perpendicular to, of course. In this case, the next step is omitted and A are directly used as the analyte in this step, point E ).

Now each circuit can be a to A and D to drawn with radius, and the points of intersection of these two circles connected to the line. The straight line is a Lot of C, that is, to a vertical line through C. The nadir point, ie the point of intersection of with is called e.

Now, the point F is determined in the same distance as the vertical E C by a circle drawn with a radius C around.

Using E and F two circles can now be constructed with radius, ( here G and H) intersect in two points. These two points define a line perpendicular to the perpendicular () to (that is, parallel to a ), which passes through the point C. This parallel is called. Thus, the first parallel is constructed.

In the same way, the parallel can be too contrived. We thus obtain a parallelogram and the intersection of the two parallels and K (all intermediate steps to the second parallel were hidden).

The track is obviously so as long as. Now only one circuit must be constructed with radius C. This cuts. Intersection L is the desired point.

Note

The proof requires that C is not on. In such a case, with the described method, a circle with the radius of an arbitrary point P are drawn, which is not on (i.e., P is used in place of the above construction C ). Then while the design is also formed a straight line (similar to in the above structure ), passing through P and is parallel. The constructed circle intersects in two points Q and R (analogous to K in the above construction ), which seems to be true that.

So that the construction can be carried out a second time, wherein the length of these two distances is used as the radius of the circle C ( i.e., P and Q are then applied as A and B in the above structure ). Here and in parallel, and since C is on, but not P, and C is not on, so the construction is now possible. It may also cause the construction of parallels in the subsequent construction will obviously skipped.

Swell

• C. Bessenrodt: Algebra I, Winter semester 2003/2004 - section 1.1 - lecture notes (PDF, 1.0 MB)