Actually, the compass that Euclid used was unable to replicate a radius if you lifted it off the paper. The compass would “collapse” under those circumstances.
And the reason you don’t lift the compass is the same reason you don’t measure with a ruler, or a protractor, to complete the constructions. It’s not about accomplishing something; it’s about how you do it and how that works logically.
As Hari pointed out, being allowed to transfer a distance shortens your constructions, but it doesn’t change what you can do. Given a line segment AB and a point C not on it, it’s easy to construct a parallelogram whose fourth vertex is D with CD parallel to AB. From there, you can draw the circle centered at C passing through D, and you have the same result that you get from picking the compass up and not collapsing it. If you have very bright students, it’s worth asking them to recreate this proof.
How does this work? How do you bisect an angle if you’re required to reset the compass if picked up. The way I learned to bisect an angle is draw an arc across the angle and then find the point of intersection of two arcs of the same radius from each of the indicated points. How do you keep the same radius for each of those arcs if you must reset?
God, I loved doing constructions. It’s probably the only segment of mathematics where I never got a wrong answer because I understood exactly what they were asking me to do. When I finished a construction I knew that I had done it right because I had proved it to myself. Doing constructions was very important to me in helping me understand the underlying principles of mathematics. I knew that the lead in the compass wasn’t exact but it was empowering to know that it was pretty damn close and was far better than a protractor. The thought and logic behind doing the construction was either right or wrong.
I think constructions can be a great teaching tool for students who have trouble grasping the underlying principles of why mathematics is so important.
The only constraint on the new arcs is that they have to intersect. There’s nothing requiring the radius of the new arcs to be exactly the same as the first one.
I learned it from a Marianist brother in 85. Never was geometry so beautiful.
Useful? In itslef, no (as with most things), but it opens your eyes to how Geometry actually works.
Assume an angle with vertex A. Using the compass, draw an arc intersecting both rays of the angle, at points B and C. Now we lift the compass, but we don’t care because it doesn’t matter what the radius of the next arc we draw is; it is only important that we draw arcs centered on B and C of the same radius.
Set the point of the compass on B, and the drawing point on C. Draw an arc through the interior of the angle. Lift the compass and put the point on C, the drawing point on B. We now have the same radius as the arc centered on B, that being the distance BC. Draw an arc in the interior of the angle, intersecting the arc centered on B at point D. Draw ray AD and you have your bisected angle.
You can most always use some variant on this to lift the compass and still have arcs of the same radius. The point to doing it this way is that your proof doesn’t have to take as an assumption that you can draw arcs with the same radii as a function of the instrument used.
I did them, although I totally forgot what the hell they were until this thread came along to remind me. However, I took my high school math courses in the form of correspondence work from Indiana University and not at my high school itself (this was an option offered to students who fucked up really bad in the math classes.) That way I could do the homework and read the textbook at whatever pace I wanted to, as long as the work was handed in at the end of the week. I have no idea if the students in the actual high school classes did them.
I absolutely LOVED geometry, but we didn’t do constructions in class. (early 80s) I think the teacher spent a little time on them and I know we were given a take-home booklet at the end of the year on them. Although I loved the class and am a geek, I didn’t do any of them (I may have looked through the booklet, but I didn’t repicate any exercise)
Brian
To expand, the (collapsing) compass and straightedge are both justified by Euclid’s postulates. A compass can draw a circle with a given radius because Euclid postulates a circle can be drawn anywhere with a given radius, and a straight edge can be used to connect to points in a line because Euclid postulates that a line can be drawn between any two points.
That a given angle can be drawn isn’t a postulate, so we can’t use a protractor.
The point of teaching kids geometry is to give them an example of using an axiomatic sytem, if we ignore the axioms in doing so, then there isn’t really much point in having the class.
(Edit: Answered above by DSYoungEsq.)
Call those indicated points Q and R. Set the radius of one intersecting arc as QR and the other as RQ. In other words, reset your compass for each arc to the distance between the two points.
This page (on the same site that Shot From Guns linked above) shows the diagram, but change steps 2 and 3 to include resetting your compass for each arc.
I learned them four or five years ago. I didn’t really care for them at the time, because I didn’t see the point. But now I’m very glad I learned them, because although I can’t draw anything freehand, I can make some really cool designs, like this. I don’t use the mathematical aspect of it much, except when I’m replicating them on the computer, when I calculate what all the relevant values are.
Ahh I see, thanks. I knew the 2nd and 3rd arcs didn’t need to be the same size as the first one, but I hadn’t figured out how to make those two equal.
