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Anonymous Poster #1

Crooked lines

10/15/2022 9:01 AM

Given a certain confined(not allowed to go outside of the defined square space),if straight lines are placed evenly apart, across the entire space, versus curved lines spaced the same distance apart,which will result in a greater total line length.It seems obvious that the curved lines will have a greater total length,but sometimes the obvious is not correct.

So,the question is:Which will have the longest total length,and how much difference?

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#1

Re: crooked lines

10/15/2022 10:31 AM

One line can totally fill the space if it's allowed to curve, since there is only one line there is no expectation of spacing....it's a loop hole in your logic....

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#2

Re: crooked lines

10/15/2022 11:23 AM

Mathematically, a line has length, but no width. Therefore, an infinite number of straight lines or curved lines could fit into a square area (2D) or a square space (3D), and the total length of both the straight lines and the curved lines would also be infinite. But is this a case in which one infinity is larger than the other, as per George Cantor? Probably not, since all of the infinite points on the curved lines are mappable on to all of the infinite points on the straight lines.

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Anonymous Poster #1
#3
In reply to #2

Re: crooked lines

10/15/2022 11:42 AM

Ok,try this:A farmer plants a field with rows either straight or wavy,like a sine wave, for instance.He is confined by structures on all sides(never mind how he got in),which rows will be longer if all the rows are equally spaced?

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#4
In reply to #3

Re: crooked lines

10/15/2022 12:01 PM

The curvy lines will be longer but fewer in number....

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Anonymous Poster #1
#5
In reply to #4

Re: crooked lines

10/15/2022 2:08 PM

Would the TOTAL length be more,less,or the same?

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#7
In reply to #5

Re: crooked lines

10/15/2022 3:31 PM

The total would be the same, to get more crops you would need to add another dimension and go vertical....

With a controlled environment, artificial lighting, CO2 you could grow your crops around the clock, your yield would increase substantially, but not necessarily cost effectively...for that you need a high money crop...

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#6

Re: crooked lines

10/15/2022 2:49 PM

Equidistant straight lines would obviously be parallel lines. Distance is measured in a direction perpendicular to the lines.

How do you define equidistant curved lines?

For example, are 1+sin(x), 2+sin(x), 3+sin(x), ... equidistant?

How about concentric circles of radius 1,2,3,...?

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#8

Re: crooked lines

10/15/2022 5:38 PM

This is a total guess...

The question is "Will curved parallel lines filling an area have a longer or shorte total length than straight lines filling the same area?"

An equivalent question is "Given parallel lines with a fixed length, how does adding curvature affect the area". If the needed area increases, then the total length to fill a fixed area would decrease.

If parallel curved lines can be constructed of circular arcs, of radius R+0.5 and R-0.5, connected with perpendicular lines (red), a median line can be drawn (green). (See figure below.) The length of the inner arc will be (R-0.5)*θ and the outer arc (R+0.5)*θ, for an average length of Rθ, which is the length of the green line.

If the lines are curved, there is more area on the convex side of the green line and less on the concave side. Because the area of a circle is proportional to the square of the radius, more area is added on the convex side than is removed on the concave side. Curving the line increases the total area, or given a fixed area, the total length would be less with curved parallel lines.

https://th.bing.com/th/id/R.f1fc7fab7ad882ac6c618a0b64f09b5e?rik=we3UaWp10SWBzQ&riu=http%3a%2f%2fweb1.kcn.jp%2fhp28ah77%2ffig8_5.gif&ehk=6pT3SKrM3ZTp%2f%2f445Bcsm0ttXJIrd3dCIpzulUeaGwA%3d&risl=&pid=ImgRaw&r=0

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#9

Re: crooked lines

10/16/2022 12:10 AM

If the lines are straight, the entire area of the square can be covered with lines.

There are lots of possible shapes that could be called "Curvy Lines", and you haven't specified whether all the curvy lines must go the full distance from one side to the other. But since you've said the lines MUST be confined within the square, for any shape of curve, there will be areas either inside, or outside, or both, the curves, where no line is permitted.

Therefore, without doing any math, I would have thought that the straight lines would have a greater total length

However, doing a test seems to have proved that wrong:

Now if you are growing crops, that additional 0.35% of length is clearly not worth the considerable extra trouble to plant and cultivate the curvy rows.

Now of course these curvy lines are only 0.1 unit apart at the points of inflection, so they aren't truly "spaced the same distance apart". The only curvy lines I can think of that could truly be "spaced the same distance apart" would be a set of concentric arcs.

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#10
In reply to #9

Re: crooked lines

10/16/2022 1:38 AM

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#12
In reply to #10

Re: crooked lines

10/16/2022 9:25 AM

Imagine a circular chart that is rotating in contact with multiple ink pens rigidly attached parallel to each other and 90 degrees tangent to the rotation center and attached to an assembly arm that is free to pivot from the center of rotation.

The assembly of pens will respond to an electrical signal.

With no signal applied,they will scribe concentric circles.

Apply a sinusoidal signal to move the pens in a sine wave.

As the chart turns, would not the lines remain equidistant and parallel to each other?

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Anonymous Poster #1
#15
In reply to #10

Re: crooked lines

10/16/2022 9:53 AM

No,like this:

Spaced so that a ball the exact diameter of the spacing would roll through the curves without binding up.

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#19
In reply to #15

Re: crooked lines

10/16/2022 5:14 PM

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#20
In reply to #19

Re: crooked lines

10/16/2022 6:28 PM

OK,you win!

I will never argue with you.

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#24
In reply to #20

Re: crooked lines

10/17/2022 1:34 PM

Oh I never argue....I might present an alternative viewpoint in the spirit of a productive debate, but argue, never....

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#25
In reply to #24

Re: crooked lines

10/17/2022 1:59 PM

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#26
In reply to #24

Re: crooked lines

10/17/2022 2:18 PM

Likewise with me.

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#35
In reply to #15

Re: crooked lines

10/18/2022 1:49 PM

The term wavy is stumping me. I wish I could understand this rotating chart scenario.

"Imagine a circular chart that is rotating in contact with multiple ink pens rigidly attached parallel to each other and 90 degrees tangent to the rotation center and attached to an assembly arm that is free to pivot from the center of rotation."

Have got a picture? Are the pens moving separately or together?

I can't see what you see.

And that other post......what ball and where is it moving?

Are you working with a rotating chart or a field?

Much more EXACT information is needed......or you see......what happens.......people start guessing and answering other scenarios.

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#36
In reply to #35

Re: crooked lines

10/18/2022 4:29 PM

Here is a circular chart recorder paper.Imagine the inking pens rigidly attached to each other and spaced a predetermined distance apart.Now apply a sine wave to the pen driver while the chart is rotating.The printed sine wave lines should be equally spaced at all points.

and pivoting from the top center of the chart,extending out to the 50 mark on the chart

As for the balls,imagine a ball that would roll freely along between the crooked lines if they are held at an angle to horizontal,with only enough clearance to allow free movement.This would insure that the lines are equally spaced everywhere,within the tolerance of the ball dimensions.

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#13
In reply to #9

Re: crooked lines

10/16/2022 9:33 AM

Yes. The deciding factor is the boundary conditions. If the shape of the boundary that makes the confined space matches a curvy line then curvy lines will be the answer. Most man-made boundaries are straight.

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#11

Re: crooked lines

10/16/2022 9:01 AM

Ask a farmer how he plants his crops, with the exception of varianceses, keep it simple.

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#14
In reply to #11

Re: crooked lines

10/16/2022 9:45 AM

I agree.

Farmers are very adept at getting the most from the available land with the lowest cost.

Farms out west are measured in square miles (640 acres per square),so even a little improvement would be beneficial and exploited to the fullest.Modern farmers are very educated and aware of new developments in technology and methods.

They(used to) set their tracked tractors (bulldozers) to run in a straight line,and get off and let them run.They start other tractors on another plot and would come back and turn them around to make another pass.There were many tractors running at once,so some farms were so large that they were constantly making the rounds.

The roads are also laid out in 1 mile grids,and sometimes they did not turn them around in time,so the road gets disked up too.

No problem,most of the secondary roads were just crude oil sprayed on the ground to bind with the soil to form a primitive asphalt.

If curved lines were more efficient,they would be exploited by now.

Even the automatic rotating sprinklers can be programmed to fill in the gap between circles.

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#16
In reply to #14

Re: crooked lines

10/16/2022 9:58 AM

640 acres is a section….

Going off topic a bit…

as far as efficiency, they see the big picture on efficiency, not only planting row crops, but harvesting… so there is give and take on it.

and now, theres a lot of equipment are control by GPS.

My brother brought a self propelled Crop Sprayer, that has GPS for spraying the crops…

The interesting g thing, when he bought the GPS attachment, that shows what’s been sprayed, what’s not, it work great and was pretty accurate. The GPS used Russian Satellites for it GPS data. I’m sure the data was probably also sent to Russian intelligence.

This past Spring, the Russian contract ended with GPS equipment supplier, and they used United States GPS satellites… it a lot less accurate…. At less for civilian use.

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#17
In reply to #16

Re: crooked lines

10/16/2022 11:53 AM

They also use satellite images to determine which areas need the most fertilizer, weed killer,etc.and apply according to the need of the particular area.

I am thinking that maybe in ground magnets or wires could be buried below the plowing depth at certain intervals to guide the modern equipment,and make them less reliant on GPS.Sort of like the wire guided robots in automotive factories follow a embedded wire in the floor.The could be installed with subsoil equipment to a depth that would not be disturbed by normal processes.This would provide excellent location data.

Key points,like end-of row and turn radius,could be designated by a binary coded array of north/south seeking reed switches or hall effect detectors.

The machines would need sensitive equipment to detect,but that is not a new technology.

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#18
In reply to #17

Re: crooked lines

10/16/2022 12:35 PM

Farming today is like ‘It not your grandfathers Oldsmobile anymore…’

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#21

Re: crooked lines

10/17/2022 2:26 AM

Unless some restrictions are placed on how the curved rows are formed, it will be possible for the combined curve lengths to exceed the combined linear lengths by an unlimited ratio.

For now, I leave this as a reader exercise. See ya later maybe.

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#40
In reply to #21

Re: crooked lines

10/18/2022 9:00 PM

This is my interpretation:

The puzzle specified "curved lines spaced the same distance apart". A bunch of sine waves or similar curves would seem to qualify for this if you measure distance in the right direction, say vertically, but not in any other direction. So I am pretty sure this doesn't qualify.

From what I could find, the only true "curved lines spaced the same distance apart" would be concentric circular arcs or curves constructed by piecing concentric arcs together. (A straight line can be considered as an arc with an infinite radius of curvature.)

A plow with a set of disks pulled through the soil of a field would generate these types of curves.

Wiki defines an equal distant line as:

"The set of the ends of equal segments laid off in a fixed direction along the normals to a given plane curve L is known as an equidistant curve of L. For example, an equidistant curve of a circle is a circle."

Equidistant Curve | Article about Equidistant Curve by The Free Dictionary

The distance between these curves would be measured as the minimum distance on a line perpendicular to both curves.

Since straight lines can be considered as arcs with infinite radius of curvature, we can look at a pair of equidistant lines and see what the effect of curvature has on the ratio of Area/total_length.

(To my surprise, the radius of curvature has no effect!)

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#22

Re: Crooked lines

10/17/2022 8:16 AM

Here's a concrete example. Consider a plow with two disks, 2 feet apart. You are going to plow a semicircle of radius 10 feet. The length of a semicircle is pi x radius. One disk will plow a semicircle of 9 feet radius and the other will plow a semicircle with 11 feet radius, for a total distance (line length) of pi x (9+11) = 20 x pi feet.

The area of a semicircle = 1/2 x pi x r2. The area plowed is the area between the two semicircles = 1/2 x pi x (112 - 92) = 20 x pi square feet.

Plow the same area in a straight line, 2 feet wide by 10 x pi feet long. The total line length is 20 x pi, the same as the total line length for the semicircle!

Note that a semicircle is not unique. The length and area scale down to any arc length. A semicircle was just used as an example.

As a practical consideration, there would be some wastage with curved lines in a rectangular field, but mathematically the length and area would be the same for straight parallel and curved parallel lines.

The reasoning in #8 is faulty, my bad...

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#23

Re: Crooked lines

10/17/2022 9:59 AM

Let the distance between the lines be 1, and, consider the "space" occupied by a single line of length L as being the the area ½ either side of it. If it is straight it will occupy an area equal to L. Now consider the same length line with "simple" curvature radius R:-

Area used by line = Θ/2π [ π (R+½)2 - π(R-½)2]

Now Θ = L/R

So area =L/2R[R2+R+¼ - (R2-R+¼)]

...........= L/2R[2R]

=L

It's the same (which has surprised me: I thought it would be bigger)

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#27
In reply to #23

Re: Crooked lines

10/17/2022 3:36 PM

It's the same (which has surprised me: I thought it would be bigger)

Yeah, it surprised me too, when I was trying to prove my original assertion that the straight lines were longer... (#24).

Cudos to AP#1, nice puzzle!

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#28
In reply to #27

Re: Crooked lines

10/17/2022 3:42 PM

I still claim the unspecified boundary condition will determine whether more lines or curves will fit a given space.

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#29
In reply to #28

Re: Crooked lines

10/17/2022 5:07 PM

I can't disagree with that. It appears that the furrow length to plowed area ratio is constant (mathematically), so the deciding factor in a practical situation would boil down to the amount of wastage of area that can't be reached with curved or straight furrows.

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#30

Re: Crooked lines

10/17/2022 10:02 PM

In the schematic below, the left side shows 4 straight lines that you can imagine as contained in a somewhat larger square. The right side shows 2 curves (sawtooth, sine, whatever you want) with a clear space between that is greater than the spacing between the lines. The length of each curve can be made arbitrarily large, so that even with only 2 the total will exceed the total length of the 4 lines.

______
______ /\/\/\/\
______
______ /\/\/\/\

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#32
In reply to #30

Re: Crooked lines

10/18/2022 1:28 AM

As I said before, there are lots of possible shapes that could be called "Curvy Lines", but in my mind, a zig-zag of straight lines (a triangular wave, if you prefer) isn't one of them.

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#38
In reply to #32

Re: Crooked lines

10/18/2022 6:49 PM

I don't think the term "curve" excludes the possibility of corners. In any event, one could imagine the sawteeth as rounded and still get unlimited length (arbitrarily close pitch). (That was the best schematic I could do with just typography.)

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#31

Re: Crooked lines

10/17/2022 11:12 PM

I'm not going to be swayed by this discussion.

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#33

Re: Crooked lines

10/18/2022 7:05 AM

How many Angels can dance on the head of a pin?

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#34
In reply to #33

Re: Crooked lines

10/18/2022 9:16 AM

As many angels that want to dance.

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#37

Re: Crooked lines

10/18/2022 5:23 PM

This doesn't apply to stripe charts and 2D fields, but spinning a line can give it relevant length/time to it's absolute length/time. WITH constant direction AND constant velocity. This is the dynamic we need for a stable clock ticker.

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#39

Re: Crooked lines

10/18/2022 7:46 PM

Straight lines. Curved lines by definition introduce a y axis value that must be included in the first and last line drawn plus the spacing. So curved lines have a chord value y <> 0 and straight lines y = 0. BUT! we are after total length, not the number of lines!

If the spacing value and the chord value are right, then it may be possible to place enough longer curved lines into the same space with the same spacing but get a longer total line length.

Hmm. Good question!

now I'm going to read the answers from the smarter people in the room!

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#41
In reply to #39

Re: Crooked lines

10/19/2022 11:25 AM

I just realized that the curved lines could be concentric circles centered in the square with the outermost complete circle tangent to the sides of the square and the innermost circle at least would have a diameter equal to the spacing rule. And then the corners could be filled with concentric arcs terminating at the sides of the square until outer concentric arcs are one space away from the vertex of the corner of the square. Add up the line lengths and they will certainly be longer than the parallel lines drawn across the square with the same spacing.

I haven't done the calculus yet but I think this would qualify as fitting the rules and would yield a longer line length total.

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#42
In reply to #41

Re: Crooked lines

10/19/2022 8:54 PM

Here are the results of your definition, using the same parameters as in my previous post:

On the right is a screenshot of the spreadsheet used to find the total length.

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#43
In reply to #42

Re: Crooked lines

10/19/2022 9:10 PM

Great visual and data.

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#45
In reply to #42

Re: Crooked lines

10/20/2022 6:45 AM

Try this alternative set of concentric circles

I replicated your result of 39.83 with the other configuration, and got this result 40.68 with the new configuration

You also get one bonus carrot in the very middle of the field.

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#46
In reply to #45

Re: Crooked lines

10/20/2022 3:22 PM

I'll try that when I can find the time...

I presume your α and β values are in radians.

I should have said that in my spreadsheet, I used the arc lengths provided by CAD.

Next time I'll use Trigonometry to actually calculate them.

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Guru

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#47
In reply to #46

Re: Crooked lines

10/21/2022 5:59 AM

Yes: radians, sorry should have included the following screen shots

This is the original configuration

You just need to change the radii to get the alternative configuration.

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#44

Re: Crooked lines

10/19/2022 9:21 PM

You are giving the problem scenario a property that is under realized. You have given the lines, both straight and curved, area, with the spacing. And using the SAME area to fill the test area.

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