"Automatic trisection" of an angle

[Qfwfq]

Draw line BG perpendicular to AB

I don't see the direction of BG as useful. Instead of points 5 onward, it is sufficient to draw the line BF and then construct two lines parallel to it, each through one of points D and E. Each of these will intersect AB in the required two points.

 

The simplest way I see to construct a parallel through a chosen point is by constructing an equilateral parallelogram. Centre the compass in the external line with more than enough aperture to reach the given line; mark one of the two intersection points and trace enough of an arc for the final step. centre the compass in the intersection and mark another point along the given line, then centre in this and intersect the first arc.

[CraigD]

Draw line BG perpendicular to AB

I don't see the direction of BG as useful. Instead of points 5 onward, it is sufficient to draw the line BF and then construct two lines parallel to it, each through one of points D and E. Each of these will intersect AB in the required two points.

That’ll work, too.

 

Which of the constructions is easiest – takes the fewest step – depends on whether you use a strict, classical “collapsing compass”, or the more liberal kind that can be used to transfer a distance – technical a divider or caliper.

 

With a collapsing compass, you must draw a perpendicular line 2 times to draw a parallel line. Drawing a perpendicular line takes 4 steps, so with a collapsing compass, my construction takes 15 steps, Qfwfq’s takes 21.

 

With a non-collapsing compass, you can draw a parallel line takes 4 or 5 steps (depending on whether you count setting the compass’s width as a step or not), so with a non-collapsing compass, my construction takes 15 steps, Qfwfq’s 13 or 15.

 

I’m still wondering about the possibility/impossibility of 5, 7, 11, etc –secting an arbitrary angle.

[Qfwfq]

With a collapsing compass, you must draw a perpendicular line 2 times to draw a parallel line.

arrrgh, splutter, I said this isn't necessary. More in detail:

 

Given a right line R and a point P external to it, set compass in P and trace a circumference C with radius sufficient to intersect R. Choose one of the two intersections A and, with the same radius, notch another point B along R. Now set the compass in B and mark the intersection Q with C (the one other than A). Points A, B, P and Q are the vertices of a parallelogram and so the line S through P and Q is parallel to R.

 

That's 3 steps unless you count the two relevant bits of C as separate steps. Considering this after posting, it had struck me that one can do it with the same compass aperture all the way through and optimize a couple of steps out, making 10 for trisection. To divide the segment in n > 3 parts, three extra steps are needed for each increment of n. Practically, 1 + 3n steps

 

I’m still wondering about the possibility/impossibility of 5, 7, 11, etc –secting an arbitrary angle.

I've never looked it up and don't find anything now but surely there have been discussions about it, through the centuries.

[Guest MacPhee]

using "compasses" to refer to one instrument is grammatically incorrect. it is not the same as "scissors".

 

Well, surely it is the same as "scissors". Scissors and compasses are the same. Two long things, with a hinge. They differ only in the way they're used. Scissors are used to cut paper, compasses to stick points in the paper. So as we say "a pair of scissors", so we should say " a pair of compasses".

 

It's particularly important that we do. Because if we say "compass", that could mean a direction-finding device utilising a magnetic needle that points to the north. In English, the word "compass" is ambiguous - it can encompass two meanings: a direction-finding device, and a pair of geometry dividers.

 

However, to resolve this ambiguity, all we have to do, is use:

 

"Compass" - to mean the magnetic needle thing;

"Pair of compasses" - to mean the geometry dividers.

 

The wonderful English language, as always, finds a way!

[Turtle]

Well, surely it is the same as "scissors". Scissors and compasses are the same. Two long things, with a hinge. They differ only in the way they're used. Scissors are used to cut paper, compasses to stick points in the paper. So as we say "a pair of scissors", so we should say " a pair of compasses".

 

It's particularly important that we do. Because if we say "compass", that could mean a direction-finding device utilising a magnetic needle that points to the north. In English, the word "compass" is ambiguous - it can encompass two meanings: a direction-finding device, and a pair of geometry dividers.

 

However, to resolve this ambiguity, all we have to do, is use:

 

"Compass" - to mean the magnetic needle thing;

"Pair of compasses" - to mean the geometry dividers.

 

The wonderful English language, as always, finds a way!

 

:doh: don't be silly macphee. we all know what kind of compass we are talking about here and until or unless common usage changes and/or dictionaries are ammended then there is simply no basis for your odd angled construction. (had to work something in there to at least give the impression of ontopictudinality ;))

 

as you were. :esmoking:

[CraigD]

With a collapsing compass, you must draw a perpendicular line 2 times to draw a parallel line.

arrrgh, splutter, I said this isn't necessary. More in detail:

 

Given a right line R and a point P external to it, set compass in P and trace a circumference C with radius sufficient to intersect R. Choose one of the two intersections A and, with the same radius, notch another point B along R. Now set the compass in B and mark the intersection Q with C (the one other than A). Points A, B, P and Q are the vertices of a parallelogram and so the line S through P and Q is parallel to R.

 

That's 3 steps unless you count the two relevant bits of C as separate steps

You’re right. Your parallelogram/parallel line construction takes 3 steps, vs. the 4 or 5 I was counting for a non-collapsing compass, or 8 for a collapsing one. I was assuming some pretty un-clever constructions.

 

The literature and folk tradition of compass construction I’ve known isn’t much concerned with how few or many steps a construction takes. Correspondingly, it’s not much concerned with collapsing vs. non-collapsing compasses, as folk as far back as Euclid had proven that any construction possible with a non-collapsing compass is possible with a collapsing one (usually with more steps) – the “Compass equivalence theorem”. It’s a fun subject, though, that’s provided me ample geekish amusement in the course of our dialog. (PS: sorry I made you splutter, Q)

 

I worked out in detail the number of steps S for n-secting a line segment using my “perpendicular” approach, C, and your “parallel” one, Q:

 

[math]S_{\text{C}}(n) = 3n +2[/math]

 

[math]S_{\text{Q}}(n) = 4n -1[/math]

 

So while Q takes fewer steps for n<3, C for for n>3. For n=3, they take the same number of steps (11).

 

A postscript on Euclid’s compass equivalence:

It wasn’t till the Late Renaissance that a clever fellow (actually 2, independently and apparently unknown to each other) proved that any compass and straightedge construction can be done not only with a collapsing compass, but without the straightedge (until checking just now, I believe that this too was known to Euclid, not a proof younger than the Calculus – hail the font of knowledge that is the internet!) – the Mohr–Mascheroni theorem – though this idea forces you to accept the rather counter-intuitive but (if you think about it for a bit) equivalent definition that a construction is a finite collection of points, rather than a collection of circles, lines, and intersections of same.

 

I’m still wondering about the possibility/impossibility of 5, 7, 11, etc –secting an arbitrary angle.

I've never looked it up and don't find anything now but surely there have been discussions about it, through the centuries.

I agree, but having not come across one in all of the internet I’ve searched so far, may be doomed to repeat one. Not a bad doom, really :)

Edited December 24, 2011 by CraigD
Corrected numbers per post #42

[Qfwfq]

Well, surely it is the same as "scissors". Scissors and compasses are the same.

This does not mean the words work the same way. Unlike the case of a scissor, each of those "two long things" would scarcely be a "same step" by itself (whereas a single blade can be used to split things in some cases),

 

[math]S_{\text{C}}(n) = 3n +4[/math]

 

[math]S_{\text{Q}}(n) = 4n -2[/math]

No. you're not optimizing, despite having acknowledged 3 steps for each parallelogram. It is enough to make the arcs sufficient while marking along the oblique line, starting from the second one. You appear to have doubled these steps in your counting.

 

[math]S_{\text{Q}}(n) = 3n + 1[/math]

[CraigD]

[math]S_{\text{C}}(n) = 3n +4[/math]

 

[math]S_{\text{Q}}(n) = 4n -2[/math]

No. you're not optimizing, despite having acknowledged 3 steps for each parallelogram. It is enough to make the arcs sufficient while marking along the oblique line, starting from the second one. You appear to have doubled these steps in your counting.

 

[math]S_{\text{Q}}(n) = 3n + 1[/math]

I think you’re forgetting that your parallel line drawing requires 3 circles, then the line, be drawn, so it’s 4 steps. One of the circles is already draws by the n-secting the drawn line steps, and you need draw only n-1 lines to n-sect the original line segment, so

[math]S_{\text{Q}}(n) = [/math]

1 (draw a line L not coincidental with given line segment AB)

+n (draw a circle C₁ at A intersecting L at X₁, a circle C₂ at X₁ intersecting L at X₂ … intersecting L at X_n)

+1 (draw a line M through B and X_n)

+3(n-1) (at the intersection of C_n-1 and M (P), draw a circle through X_n-1 intersecting M at Q, draw a circle at Q through P intersecting C_n-1 at R, draw a line – “the parallel” – through X_n-1 and R, etc.)

= 4n -1 (my first post’s [imath]S_{\text{Q}}(n) = 4n -2[/imath] was wrong by 1)

 

[math]S_{\text{C}}(n) = [/math]

n-1 (draw a circle at B through A intersecting AB at X₂ … at X_n-1 interstecting AB at X_n)

+n (draw a circle C₁ at A through B, a circle C₂ at A through X₂ … a circle C_n at A though X_n-1)

+1 (draw a circle D at X₂ through A)

+1 (draw a line through the intersection of C₂ and R – “the perpendicular”)

+2 (draw line E and F through A and each of the intersections of C_n and R)

+(n-1) (draw a line through the intersection of C₁ and E and F, … through the intersection of C_n-1 and E and F)

= 3n +2 (my first post’s [imath]S_{\text{C}}(n) = 3n +4[/imath] was also wrong by 2)

 

This is hard to visualize from our various descriptions, so here're a couple of sloppy sketches made with MSPaint (the only thing I have handy at the moment, bad as it is):

[Turtle]

This is hard to visualize from our various descriptions, so here're a couple of sloppy sketches made with MSPaint (the only thing I have handy at the moment, bad as it is):

 

 

i'm still trying to visualize these with the drawings. :blink: :lol: thnx for all the work. :thumbs_up: (U 2 Q) i see in drawing 2 that the resulting parallel lines have incident points on a vesica piscis. is this just incidental (;) :doh:), or could one say the construction of them is "on a vesica piscis"?

[CraigD]

i see in drawing 2 that the resulting parallel lines have incident points on a vesica piscis. is this just incidental (;) :doh:), or could one say the construction of them is "on a vesica piscis"?

Both constructions make vesica pisces. Both start by constructing 3 or 2 line segments of equal length, each time constructing a VP. The VPs in the Q sketch are harder to pick out because of lots of nearby circles of the same radius that aren’t VPs (in the Q construction, all of the circles have the same radius, and there are a lot of them).

 

You might sensibly say, I think, that equal length line segments constuctions are “on vesica pisces”

 

The VP is arguable the most terse compass construction there is, so I’m not surprised. You might not imagine it appears in constructions so often, because traditionally, we don’t sketch entire circles, only arcs near their intersections with lines and each other. Since I made my sketches with unfriendly, bare-bones, freebie MSPaint, I just copied the same circle and pasted it different places (you’ll notice, not always quite the right ones :)), so all the circles are fully drawn, and VPs pop out at you visually.

 

Strictly, the only allowed operation with a compass in a classical construction is to draw a complete circle – that is, you can’t use where you stop or start sketching a partial arc in your construction as useful points in the construction, only skip actually sketching a portions of the circle to save time or make your sketch less cluttered-looking. Were it not for this inevitable drafting tradition (it’s human nature to try to avoid visual confusion), most sketches of constructions would be cluttered, but have a lot of interesting coincidental figures, like VPs.

[Qfwfq]

I think you’re forgetting that your parallel line drawing requires 3 circles, then the line, be drawn, so it’s 4 steps.

No, I'm not forgetting that, I think you’re forgetting that adjoining parallelograms share a side. ;)

[Jayeskay]

This link may be of general interest: <http://www.zolkorp.com/TRiSeK.htm >.

[phision]

This link may be of general interest: <http://www.zolkorp.com/TRiSeK.htm >.

This link didn't work for me!

 

If you have a radius of a circle on your compass and position this along-side your straight edge, so as to locate the centre of the circle you wish to draw does this go against the spirt of the problem?

[Jayeskay]

This link didn't work for me! ......

 

I copy-pasted it from the link itself, which usually works. This should take you.

< http://www.zolkorp.com/TRiSeK.html> .

[Qfwfq]

This link may be of general interest: <http://www.zolkorp.com/TRiSeK.htm >.

I've no doubt that you trisect an angle, only not with a SE&C method.

[Jayeskay]

I've no doubt that you trisect an angle, only not with a SE&C method.

 

 

Looking at the three diagrams on < http://users.tpg.com.au/musodata/trisection/trisecting_any_angle.htm >,

what method other than SE&C is being used?

[phision]

I’m still wondering about the possibility/impossibility of 5, 7, 11, etc –secting an arbitrary angle.

Was looking for diagrams and found this about odd numbered integers http://commonsensequantum.blogspot.com/2011/03/archimedes-angle-trisection-or-tripling.html

 

< http://www.zolkorp.com/TRiSeK.html> .

I thought the wee trisecting machine on this link was brilliant. :wave:

 

I've no doubt that you trisect an angle, only not with a SE&C method.

 

On this link, http://en.wikipedia....ngle_trisection , in section 4.4 With a marked ruler , it appears to me that instead of using a marked ruler, you could use the compass set to radius AB to construct the line AD, so that point C(on the circle), is the same distance from D, as A is from B using your straight edge to make the points A, C and D co-linear.

 

You could then go on to use the compass still set to radius AB and centred at point D, to create a circle going through point C. This circle would intersect the original circle at another point which would be the reflection of point C along line BD, calling this point E. If a line is drawn from E through B and extended beyond this it would divide angle a, into angle b and c.

 

This method would be good for an angle a of up to 90°. Angles of more than this could be bisected one or two times before doing this in order to get to the trisected angle!