MEOW #153 Viscious Circle

[DrTom]

Is anyone going to complain that half of those points of a circle aren't actually 4 away, but are 3 x square root of 2 or roughly 4.2426 away?

Yeah, didn't think so.

You know... if you had done a distance of 5, ... well, think about it.

Well actually they aren't, and the points of a circle if measured from the origin are all exactly the same distance. Note this circle with a R of 4 from an online drawing app:

Cardinal points (N,S,E,W) are all a radius of 4, and that is easily seen by counting the 4 squares, BUT the Radius in not in squares it is in distance and when one rotates it around the midpoint it is virtually impossible for the distances all to be 4 "squares", for that to happen the circle would have to be a square (as shown) and hardly applicable in a FULL MOON puzzle. However, if you were to measure the actual distance from the center to any point on the circle you would find it the same, which it HAS to be since it is drawn by circumscribing arcs a fixed distance from the center. The clue could not say "Location for a 3.36 cm radius" = ARM.

The "ordinal" points are not 4.246 away if the cardinal ones are 4, if anything the "non-ordinal" points are LESS than 4 squares away, but again we are not talking squares we are talking distance. I'm sure there is a math major out there who can explain it better, I just know what my process was and that the "letters" for FULL MOON were all where they should be. Had it been 5 the answer woudl have been OAAI LOLR and I can't even ROT-10 that to an answer.

[boharr]

^WOW

[MatthewL]

Come for the meta, stay for the geometry lesson. Up next, Dr. Tom explains the Pythagorean Theorem.

[woozy]

But the center of the circle is the *center* of the square. And then it is exactly four distance to the *centers* of the four cardinal squares.

The circle doesn't hit the center of any other squares but it's pretty close (for some values of "close") for the four NE,NW,SW,SE squares. The remaining sixteen squares the circle intersects are far from the center and most wouldn't think they count.

But this always bugged me a little bit.

(Watch this space...)

EDIT: se what I was getting at is that if the X is centered at (0,0) and the F at (0,4) than the U is centered at (3,3). The distance from (0,0) to (3,3) is 3 times the square root of 2 (or the square root of 18) which is not equal to 4. I'm not quite sure what you thought I was trying to say.

[HeadinHome]

Haven’t looked at the “shove” but got it now. The 15A one distracted more than helped me, and ditto 61D (long time looking at “RAIO” based on those two clues. All makes sense in retrospect, of COURSE! Now to look a the shove….

You'll have to explain that to me after the reveal, I am not sure how you got that?

I used the 45-degree thing, and the fo(u)r clue, to start at B (center) and go 45 degrees (diagonally) out, four spaces, to the letters in four corners that far away.

[DrTom]

But the center of the circle is the *center* of the square. And then it is exactly four distance to the *centers* of the four cardinal squares.

The circle doesn't hit the center of any other squares but it's pretty close (for some values of "close") for the four NE,NW,SW,SE squares. The remaining sixteen squares the circle intersects are far from the center and most wouldn't think they count.

But this always bugged me a little bit.

(Watch this space...)

EDIT: se what I was getting at is that if the X is centered at (0,0) and the F at (0,4) than the U is centered at (3,3). The distance from (0,0) to (3,3) is 3 times the square root of 2 (or the square root of 18) which is not equal to 4. I'm not quite sure what you thought I was trying to say.

Well you are going to have to PM Archimedes or Euclid because they came up with the geometry of a circle, not me. I will say that I took an online ruler and checked the radius to any point (well about 20 points) on the circle and it was exactly the same every time. I also imported the picture I put on the forum into Excel and drew a straight line from the origin to a cardinal point. Then I rotated that line around by 15 degrees and it was always the same length.

I learned this lesson the hard way about 38 years ago when I installed a ceiling fan in my first house. “Cut a 4 inch hole” the instructions said, so I did. It was only when I put the round mounting plate up and it looked like this:

that I realized they meant a 4 inch ROUND hole.

Surely there is a mathematician out there who can explain it to us?

T

[woozy]

Well you are going to have to PM Archimedes or Euclid because they came up with the geometry of a circle, not me. I will say that I took an online ruler and checked the radius to any point (well about 20 points) on the circle and it was exactly the same every time. I also imported the picture I put on the forum into Excel and drew a straight line from the origin to a cardinal point. Then I rotated that line around by 15 degrees and it was always the same length.

I learned this lesson the hard way about 38 years ago when I installed a ceiling fan in my first house. “Cut a 4 inch hole” the instructions said, so I did. It was only when I put the round mounting plate up and it looked like this:
ceiling fan.jpg
that I realized they meant a 4 inch ROUND hole.

Surely there is a mathematician out there who can explain it to us?

T

Yes, and you drew the circle fabulously. But notice that although the circle crosses many squares it doesn't cross the dead center of the squares.

See if we measure from dead center of the B and we assume the squares are 1 by 1. then the north east corner of the square with the B and the SW corner of the square with the will but the square root of 1/2 away. (That is because the east edge of the square with the B is 1/2 away and the north edge of the square with a B is 1/2 away. So the path from the center to the corner is the third side of a right triangle with legs that are 1/2 the squares distance. So by the Pythogorean Theorem/Distance Theorem the square of the path from the center to the corner is (1/2)squared + (1/2)squared or 1/4 + 1/4 or 1/2, and so the path is the square root of 1/2). We do a similar calculations to get the center of the square with the L is square root of 2 from the center (because is is 1 to the east and 1 to the north of the center so the hypotenuse is square root of 1 squared plus 1 squared). And so on. Finally, the square with the U. The SW corner is 2.5 east and 2.5 north of the center so the *CORNER* is square root of 2.5 squared + square root of 2.5 squared or, more simply 2.5 times the square root of 2 or about 3.5. The *center* of the square with the U is 3 east of the center of the circle and three north of center of the circle. So using the distance formula or Pythogorean theorem it is the square root of 3^2 plue 3^2 or square root of 18 or 3 times square root of 2 (alternatively more simply it is three squares away and the diagonal of a square is square root of 2 so it is 3 times square root of 2 away). That is about 4.26. So you have the *corner* being 3.53 and the *center* being 4.26 and circle being in the square somewhere in between but closer to the center than to the corner. But it does not go through to center.

Hmm, if I had those instructions I'd have probably tried to make a hole that had an *AREA* of 4 which would been only 2.2567583341910251477923178062431 inches across.

Common sense is just not meant for the likes of you and me.....

This is probably a really *awful* promotion for my latest meta (https://crosshare.org/crosswords/MToNVE ... re-squared) which is on this theme. (My next meta will be about root canals and colonoscopies)

[boharr]

Is The XWord Rabbit listening?

[DrTom]

Woozy, I assume that Meta will be titled; “You’re either down my throat or up…”

[DrTom]

Well you are going to have to PM Archimedes or Euclid because they came up with the geometry of a circle, not me. I will say that I took an online ruler and checked the radius to any point (well about 20 points) on the circle and it was exactly the same every time. I also imported the picture I put on the forum into Excel and drew a straight line from the origin to a cardinal point. Then I rotated that line around by 15 degrees and it was always the same length.

I learned this lesson the hard way about 38 years ago when I installed a ceiling fan in my first house. “Cut a 4 inch hole” the instructions said, so I did. It was only when I put the round mounting plate up and it looked like this:
ceiling fan.jpg
that I realized they meant a 4 inch ROUND hole.

Surely there is a mathematician out there who can explain it to us?

T

Yes, and you drew the circle fabulously. But notice that although the circle crosses many squares it doesn't cross the dead center of the squares.

overlyanal.png

See if we measure from dead center of the B and we assume the squares are 1 by 1. then the north east corner of the square with the B and the SW corner of the square with the will but the square root of 1/2 away. (That is because the east edge of the square with the B is 1/2 away and the north edge of the square with a B is 1/2 away. So the path from the center to the corner is the third side of a right triangle with legs that are 1/2 the squares distance. So by the Pythogorean Theorem/Distance Theorem the square of the path from the center to the corner is (1/2)squared + (1/2)squared or 1/4 + 1/4 or 1/2, and so the path is the square root of 1/2). We do a similar calculations to get the center of the square with the L is square root of 2 from the center (because is is 1 to the east and 1 to the north of the center so the hypotenuse is square root of 1 squared plus 1 squared). And so on. Finally, the square with the U. The SW corner is 2.5 east and 2.5 north of the center so the *CORNER* is square root of 2.5 squared + square root of 2.5 squared or, more simply 2.5 times the square root of 2 or about 3.5. The *center* of the square with the U is 3 east of the center of the circle and three north of center of the circle. So using the distance formula or Pythogorean theorem it is the square root of 3^2 plue 3^2 or square root of 18 or 3 times square root of 2 (alternatively more simply it is three squares away and the diagonal of a square is square root of 2 so it is 3 times square root of 2 away). That is about 4.26. So you have the *corner* being 3.53 and the *center* being 4.26 and circle being in the square somewhere in between but closer to the center than to the corner. But it does not go through to center.

Hmm, if I had those instructions I'd have probably tried to make a hole that had an *AREA* of 4 which would been only 2.2567583341910251477923178062431 inches across.

Common sense is just not meant for the likes of you and me.....

This is probably a really *awful* promotion for my latest meta (https://crosshare.org/crosswords/MToNVE ... re-squared) which is on this theme. (My next meta will be about root canals and colonoscopies)

Like I said, PM Archimedes or Euclid cause you lost me.

[woozy]

Woozy, I assume that Meta will be titled; “You’re either down my throat or up…”

Damn it. Now I'm thinking I might actually find myself thinking I actually *will* have to make that meta...

[DrTom]

Woozy, I assume that Meta will be titled; “You’re either down my throat or up…”

Damn it. Now I'm thinking I might actually find myself thinking I actually *will* have to make that meta...