Compass is an essential tool to learn mathematics in elementary to junior high school, in order to draw various figures. Students are often asked to draw actually figures in a given construction, using a compass and ruler. If one has left his compass at home, he will be scolded at by the teacher, or lose in examination, etc. When I was at such an age, I had two sets of compass, one was always in the bag to school and the other I used at home to study. For a ruler one could use the edge of a notebook or even that of a pencil, if he doesn’t have one, but there is almost no other method that will play the role of a compass.

In japanese hight school, students are asked to solve geometric problems more analytically, using linear equation, trigonometric function and integration, so a compass comes to be less needed. In university, one might use one in practice of technical drawing, but since recent drawing mostly is through CAD, I don’t know how often a compass used there.

Anyway, in the lecture of geometry which I take, the way of drawing figures with compass and ruler was discussed again.
Through my eyes of today, not as a student who needed to study to graduate school or enter to an upper education, but as an adult purely enjoying to get a knowledge, it is a wonder that straight lines and right angles can be created by a tool for drawing circles.

A bisection, a line which vertically divide other line into half, is created by joining both intersections of two arcs of the same radius, each of whose centers are set on the each terminal of the line to be divided.
Bisection from two circle(arc)s drawn by a compass

A bisection of an angle. First, draw an arc, so that its center is at the vertex and it crossed the both lines which produce the angle. Then draw two arcs with the same radious from the both intersections of the first arc and the lines as mentioned. Join the intersection of the latter two arcs and the point of the angle.
Bisection  of an angle from three circle(arc)s drawn by a compass

Even a parallel pair of lines can be drawn from the three circle(arc)s with the same radius, by a compass.
Parallel lines from three circle(arc)s drawn by a compass
Assume that the line below is given. First, draw a circle from a point anywhere on the given line, so that the arc crosses it. Then from the intersection draw the second circle to cross the first circle. Finally, draw the third circle from the intersection of the former two circles, to cross the second circle. The parallel line is drawn by joining the intersections of first/second circles and second/third circles.
This method is based on drawing a rhombus, that is, a square all whose four sides have the same length, which corresponds to the radius of the circles.
Creating a rhombus with three circles

Later I found the method to draw any figures with a compass and a ruler is a well-known problem of geometry, which I would like to call “mathematician’s dream”. It’s surprising, a compass is such a creative tool.


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