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Topic: Straight Circles. (Read 1181 times) |
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rloginunix
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Straight Circles.
« on: Jul 2nd, 2014, 9:23am » |
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Straight Circles.
Draw a straight line using circles only.
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rloginunix
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Re: Straight Circles.
« Reply #2 on: Jul 2nd, 2014, 10:58am » |
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Nice try, towr.
No.
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Grimbal
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Re: Straight Circles.
« Reply #3 on: Jul 2nd, 2014, 12:37pm » |
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That means you can't draw a straight line. Doesn't it?
Or do you mean to draw an arbitrary but finite number of points that are all on a straight line?
Or can I use the diameter? 
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| « Last Edit: Jul 2nd, 2014, 12:44pm by Grimbal » |
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rloginunix
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Re: Straight Circles.
« Reply #4 on: Jul 2nd, 2014, 1:02pm » |
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You guys are super smart, I've got to be careful here.
Sorry if I'm being too vague but let's see if we can zero in on a solution.
I know that you know the information needed to solve this one. It's simpler than you think.
Don't want to inflict my way of thinking on anyone but my hint 1 would be divide the problem into pieces. How many circles do you need to draw one straight line segment using circles only?
Hint 2: how do you convert between circular and linear motions?
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rloginunix
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Re: Straight Circles.
« Reply #5 on: Jul 2nd, 2014, 1:20pm » |
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Trying to scale the quality of the clues on the fly here, please don't kill me. I've put this puzzle together simply by withholding some amount of "front-end" information and exposing the tail end of it.
May be the verb "draw" should be replaced with "produced". But then again, you can actually do it in the real world. Well, let's see how it goes.
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rloginunix
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Re: Straight Circles.
« Reply #6 on: Jul 2nd, 2014, 1:22pm » |
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A hint for Grimbal: your last question is right on the money.
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alien2
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Re: Straight Circles.
« Reply #7 on: Jul 2nd, 2014, 2:10pm » |
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A circle could be a straight line if you apply 3D technical drawing.
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dudiobugtron
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Re: Straight Circles.
« Reply #8 on: Jul 2nd, 2014, 4:37pm » |
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I haven't read the clues, so this may be already discounted, but here are my attempts:
1) The 'obvious' solution (or, perhaps, the obviously incorrect solution) is to use 3 dimensions to manipulate the circle. Eg; view it from the side after rotating it 90 degrees about the y axis. Or, view it in a 2d plane wrapped in 3d around a sphere that the circle is a great circle of.
2) you can use two overlapping, congruent circles to define two points. You can then draw arbitrarily many pairs of overlapping congruent circles with centre at these two points to find arbitrarily many points along the perpendicular bisector, which is a line. This satisfies Grimbal's conjectured requirement.
3) Cover the outer rim of the circle with ink or paint, and roll it away from you in a straight line. A hoola hoop would be my circle of choice for this.
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dudiobugtron
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Re: Straight Circles.
« Reply #9 on: Jul 2nd, 2014, 4:53pm » |
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Ok, having now read the hidden clues, it looks like my 2nd answer is the one that is in the correct vein. I like the 3rd answer best though. 
PS: for those who are interested in geometrical construction puzzles, search the internet for 'Euclid the game'. It's a pretty fun game where you have to construct stuff using a ruler and compass, but it automates enough of it for you that you can focus on solving the puzzle. The first 10 levels are just teaching you how to play the game and use all of the construction techniques; level 11 is where the game starts to give you puzzles to solve.
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rloginunix
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Re: Straight Circles.
« Reply #10 on: Jul 2nd, 2014, 7:57pm » |
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Thank you for an interesting observation, Alien2, but you don't have to step out into the third dimension to solve this problem. There are no tricky perspective or projection effects, just a pure two D math.
dudiobugtron, I like your 3-rd idea, MacGyver style, but I have something different in mind (even though it somehow echoes with the intended solution). Your second idea does produce collinear points, as I discussed it in Compass Only Pentagon Constructions, but it doesn't produce a solid line.
As I noted before, Grimbal came very close to the intended solution. In the end what you get is exactly the diameter of a circle - as a solid line. However, this is not a Euclidean ruler and compass problem.
If I may direct your attention to the problem statement you will see that I chose my words carefully. I didn't say "using a compass only". I said "using circles only".
As a consequence of this - a hint: you are free to use, say, a marked ruler to set some parameters of a circle (or circles). But once you set them you can only use circles to draw the line and nothing else (well, you've got to use a pencil of course).
I hope I didn't give much away and kept it interesting.
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towr
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Re: Straight Circles.
« Reply #11 on: Jul 2nd, 2014, 10:21pm » |
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Fold a circle in half, then use a circle made of graphite* to draw a straight line along the diameter of the folded circle?
(*I'm too pedantic to use a pencil when the problem says to use only circles)
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rloginunix
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Re: Straight Circles.
« Reply #12 on: Jul 3rd, 2014, 8:35am » |
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I thought that with the drawing problems a mark producing instrument like a pen or a pencil comes by default. Apologies if I was wrong, I'll be more explicit next time.
I do have to praise your inventiveness, towr.
I guess MacGyverish solutions can be considered as solutions (albeit not the intended ones). dudiobugtron's idea 3 may be considered but it doesn't guarantee a straight line, it could be wavy.
towr's suggestion guarantees a straight line but the intended solution is purely mathematical and the line is guaranteed to be absolutely straight. It has to do with motion, plane curves ....
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Grimbal
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Re: Straight Circles.
« Reply #13 on: Jul 3rd, 2014, 1:21pm » |
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I think I got it....
You need 2 circles, one half the size of the other.
Right?
Another idea I had: the horizon is a circle. Yet it is a straight line*. It is just a matter of point of view.
* Well, almost.
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| « Last Edit: Jul 3rd, 2014, 1:21pm by Grimbal » |
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rloginunix
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Re: Straight Circles.
« Reply #14 on: Jul 3rd, 2014, 4:35pm » |
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Yes, Grimbal. You got it and you are absolutely right!
The intended solution: a degenerate hypocycloid - a path traced by a point (pencil) on a circle of radius R rolling without slipping inside the fixed circle or radius 2R which is exactly 2R's diameter which is a straight line segment. Duplicate it again and again and you can draw a straight line of an arbitrary length, infinite if you want (awkward but doable). In US they sell a toy set called "spyrograph" that deals with these things.
I hope its clear now why I replied "no" to towr's original query - I figured if I said "yes" it would steer you away from the intended solution.
I also see now that the answer I was looking for is probably way too specific. If you didn't study these things before there is no way to deduce them. I should've added more clues to the problem statement.
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dudiobugtron
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Re: Straight Circles.
« Reply #15 on: Jul 3rd, 2014, 5:14pm » |
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That is an awesome answer, and very satisfying. I never would have got it though, even with Grimbal's hint, so I don't feel bad for looking.
on Jul 3rd, 2014, 8:35am, rloginunix wrote:| I guess MacGyverish solutions can be considered as solutions (albeit not the intended ones). dudiobugtron's idea 3 may be considered but it doesn't guarantee a straight line, it could be wavy. |
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If you spin the circle sufficiently before you let go of it, then you will get splattered with paint, but the line it draws will be straight enough for all practical purposes (otherwise, bicycles (and particularly unicycles) wouldn't work very well).
A cool way to see this is action is to spin a hoola hoop (so the top moves toward you) with a flick of your wrist as you throw it away from you. Once it lands, it will bounce a bit, and then come right back to you in a straight line. You can throw it a significant distance once you get good at it; as long as it is spinning fast enough it won't fall over or change direction.
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towr
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Re: Straight Circles.
« Reply #16 on: Jul 3rd, 2014, 9:54pm » |
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I thought of spirograph after your last hint (they sell it over here as well), but there's a reason those have 'teeth', because there's practically no way to get traction with plain circles.
Quote:| I also see now that the answer I was looking for is probably way too specific. If you didn't study these things before there is no way to deduce them. I should've added more clues to the problem statement. |
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I think it's fine to just add hints to the thread as needed while people are making guesses.
[e]fixed spelling[/e]
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| « Last Edit: Jul 4th, 2014, 6:37am by towr » |
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rmsgrey
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Re: Straight Circles.
« Reply #17 on: Jul 4th, 2014, 4:31am » |
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The official version of the toy is called Spirograph - there may be a knock-off version out there, but if so it's strictly unofficial...
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rloginunix
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Re: Straight Circles.
« Reply #18 on: Jul 4th, 2014, 12:40pm » |
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towr's and rmsgrey's toy spelling makes more sense actually. The one I saw was likely used to avoid litigation (and paying royalties).
dudiobugtron, I take your point about the spinning hoola hoop. I tested this problem on my kids and their suggestion was to insert the pencil into the center of the disc and roll it on the floor along the wall - the straight line will be traced on the wall since R remains the same.
You can use GeoGebra or Cinderella to create your own hypocycloid (and other) animations. Cinderella can do all kinds of them, it's pretty awesome software.
Not sure if you will accept it but it just occurred to me that potentially there's yet another solution for the recent "Compass Only Regular Pentagon Construction" problem: you simply roll without slipping a unit circle inside the fixed 5-circle, where 5-circle is a circle with the radius 5. The unit circle will then mark the 5 vertexes of the regular pentagon.
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Lupin
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Re: Straight Circles.
« Reply #19 on: Jul 5th, 2014, 3:54pm » |
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How about you take a regular ball. Put your pen on its surface and turn the ball 360°. You have drawn a straight line, but it being the circumference of the ball, it is a circle. Or is it the other way around? Like, get a roller, put a line of paint on the middle and roll it over a surface – that would be a straight line drawn with a circle (the circumference of the cylinder). How about dipping a pizza cuter in ink and drawing a line on a paper?
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Grimbal
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Re: Straight Circles.
« Reply #20 on: Jul 7th, 2014, 6:39am » |
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Right. A straight line but in spherical geometry.
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Annettagiles
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Re: Straight Circles.
« Reply #21 on: Sep 25th, 2014, 4:05am » |
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It is really funny and stupidous as no one can't do this.
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