| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 542.1 | X marks the spot. | CHOVAX::YOUNG | Chi-Square | Wed Jul 23 1986 20:36 | 25 |
|
+---+---+---+---+---+---+---+---+
| | * | | | * | | | |
+---+---+---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
| | | | | | X | | |
+---+---+---+---+---+---+---+---+
| * | | | * | | | | |
+---+---+---+---+---+---+---+---+
| | | | | | | * | |
+---+---o---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
| | | | * | | | | |
+---+---+---+---+---+---+---+---+
| | | | | | | * | |
+---+---+---+---+---+---+---+---+
How this Lynn?
-- Barry
|
| 542.2 | re. .0 on Problem 1 | THEBUS::KOSTAS | Kostas G. Gavrielidis <o.o> | Wed Jul 23 1986 20:38 | 28 |
| re. .0
> Problem 1: Find another point in the diagram which has the same properties
> as 'o'.
How's this?
+---+---+---+---+---+---+---+---+
| | * | | | * | | | |
+---+---+---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---o---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
| * | | | * | | | | |
+---+---+---+---+---+---+---+---+
| | | | | | | * | |
+---+---o---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
| | | | * | | | | |
+---+---+---+---+---+---+---+---+
| | | | | | | * | |
+---+---+---+---+---+---+---+---+
Kostas G.
|
| 542.3 | 1 answer to #2 | CHOVAX::YOUNG | Chi-Square | Wed Jul 23 1986 20:43 | 13 |
| #2 :
The cartesian plane, with only integer vertices distinguished, has
an infinite number of such rotation points.
I guess you where thinking finite though, huh?
Also, did you intend that +90, -90 degrees be the only rotations
allowed?
-- Barry
|
| 542.4 | | CLT::GILBERT | It's a Dusey | Wed Jul 23 1986 23:49 | 1 |
| re .2 Sorry, Barry's got it right.
|
| 542.5 | | CLT::GILBERT | It's a Dusey | Thu Jul 24 1986 00:23 | 12 |
| 542.6 | Look a little deeper... | MODEL::YARBROUGH | | Thu Jul 24 1986 12:36 | 13 |
| re .2: close, but no cigar. See .1 for the correct solution.
Since we are talking about square tiles being rotated, clearly only
multiples of 90 deg. are relevant.
Notice that, for the two centers of rotation in note .1 that
the squares that rotate into one another are completely different.
The fact that there are seven squares involved is relevant. To see
this more clearly, assign letters A-G to the distinguished tiles
and write down the pairs that are separated by 90 deg in each case.
Also note that the radii are different in the two cases.
Lynn
|
| 542.7 | | CLT::GILBERT | It's a Dusey | Thu Jul 24 1986 14:54 | 31 |
| 542.8 | 8 rotations, 7 points | GALLO::JMUNZER | | Thu Jul 24 1986 15:59 | 35 |
| Seven points seem to fit naturally:
+---+---+---+---+---+---+---+---+
| | C | | | E | | | |
+---+---+---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
| | | | | | X | | |
+---+---+---+---+---+---+---+---+
| G | | | A | | | | |
+---+---+---+---+---+---+---+---+
| | | | | | | D | |
+---+---o---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
| | | | B | | | | |
+---+---+---+---+---+---+---+---+
| | | | | | | F | |
+---+---+---+---+---+---+---+---+
If R means rotation of 90 degrees about X, and
if r means rotation of 90 degrees about o, then
r R r R
A -----> B -----> C -----> D -----> A
and
R r R r
A -----> E -----> F -----> G -----> A
Seven points, each connected to another point by an R or an r.
Part of the difficulty of Problem #2 is that a third rotation can't fit
connections together so easily.
John
|
| 542.9 | four, not seven | MODEL::YARBROUGH | | Thu Jul 24 1986 16:27 | 6 |
| As pointed out in .-1, there are two cycles of four tiles each which
satisfy the rotational condition independently. So in fact seven
is not the relevant number, but four is.
My current suspicion is that three centers of rotation are possible,
but not in the 8x8 confines; I think a larger space might work out.
|
| 542.10 | No, seven is irrelevant | CLT::GILBERT | It's a Dusey | Thu Jul 24 1986 17:15 | 31 |
| Find the two centers.
+---+---+---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
| | X | X | | X | | | |
+---+---+---+---+---+---+---+---+
| | | | | X | | X | |
+---+---+---+---+---+---+---+---+
| X | X | | | | | X | |
+---+---+---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
Find the two centers.
+---+---+---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
| | | | X | X | | | |
+---+---+---+---+---+---+---+---+
| | | | | X | | X | |
+---+---+---+---+---+---+---+---+
| | | X | X | | | X | |
+---+---+---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
P.S. I wonder whether whether I can prove three centers are impossible
before Lynn finds an example. :-).
|
| 542.11 | Party of the first part... | CHOVAX::YOUNG | Chi-Square | Thu Jul 24 1986 21:48 | 19 |
|
+---+---+---+---+---+---+---+---+
| | | | 1 | | | | |
+---+---+---+---+---+---+---+---+
| | X | X | | X | | | |
+---+---+---+---+---+---+---+---+
| | | | | X | | X | |
+---+---+---+---+---+---+---+---+
| X | X | | | | | X | |
+---+---+---+---+---+---+---+---+
| | | | 2 | | | | |
+---+---+---+---+---+---+---+---+
How's this?
-- Barry
|
| 542.12 | Party of the second part. | CHOVAX::YOUNG | Chi-Square | Thu Jul 24 1986 22:00 | 19 |
| Here we go:
+---+---+---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
| | | | X |<X>| | | |
+---+---+---+---+---+---+---+---+
| | | | | X | | X | |
+---+---+---+---+---+---+---+---+
| | | X | X |<2>| | X | |
+---+---+---+---+---+---+---+---+
| | | | | | | | |
+---+---+---+---+---+---+---+---+
Note : <1> is on an 'X'.
-- Barry
|
| 542.13 | | CLT::GILBERT | It's a Dusey | Fri Jul 25 1986 13:59 | 69 |
| 542.14 | | GALLO::JMUNZER | | Fri Jul 25 1986 21:37 | 24 |
| Three centers of rotation:
+---+---+---+---+---+---+---+---+---+
| | * | | * | | * | | * | |
+---+---+---+---+---+---+---+---+---+
| * | | * | | * | | * | | * |
+---+---+---+---+---+---+---+---+---+
| | * | | * | | * | | * | |
+---+---+---+---+---+---+---+---+---+
| * | | * | o | * | o | * | | * |
+---+---+---+---+---+---+---+---+---+
| | * | | * | o | * | | * | |
+---+---+---+---+---+---+---+---+---+
| | | | | * | | | | |
+---+---+---+---+---+---+---+---+---+
| | | | * | | * | | | |
+---+---+---+---+---+---+---+---+---+
| | | * | | * | | * | | |
+---+---+---+---+---+---+---+---+---+
| | | | * | | * | | | |
+---+---+---+---+---+---+---+---+---+
John
|
| 542.15 | WOW. | CHOVAX::YOUNG | Chi-Square | Sat Jul 26 1986 00:48 | 9 |
|
You know, I would have bet anything that three centers of rotation
could not exist.
VERY nice.
-- Barry
|
| 542.16 | | CLT::GILBERT | like an eager child | Sun Jul 27 1986 01:46 | 1 |
| Oh, Wow! Amaze us again.
|
| 542.17 | | CLT::GILBERT | It's a Dusey | Mon Jul 28 1986 18:13 | 28 |
| Somewhere this argument is flawed...
Three centers cannot be in a row.
Assume we can have three centers in a row, at points (a,0), (b,0) and (c,0)
(with a < b < c), let x0 and x0 be the x-coordinates of the (marked) tiles
having least and greatest x-coordinate, respectively. Let y0 be the
y-coordinate of the tile having the greatest y-coordinate. By symmetry, if
for each marked tile (x,y), we also mark the tile (x,-y), we will also have
a solution, so we have -y0 as the least y-coordinate.
(x0,y0) +-------------------------------+ (x1,y0)
| |
| |
| |
| A B C |
| |
| |
| |
(x0,-y0) +-------------------------------+ (x1,-y0)
Now if point A is able to rotate a right-most tile somewhere into the
rectangle, we must have 'x1 <= a+y0'. For point B, we have 'y0 <= x1-b'.
But
y0 <= x1-b <= (a+y0)-b implies b <= a.
Note that we haven't even considered point C, so we might reasonably
conclude that *two* points cannot be in a row!
|
| 542.18 | >, not <. | MODEL::YARBROUGH | | Mon Jul 28 1986 20:29 | 2 |
| 'x1 <= a+y0' is incorrect; should read
'x1 >= a+y0'.
|
| 542.19 | | CLT::GILBERT | It's a Dusey | Mon Jul 28 1986 21:29 | 3 |
| No, that's not it. The equation should put an upper bound on x1;
if x1 is *too* large, there's no way point A can rotate it into
the rectangle.
|
| 542.20 | | GALLO::JMUNZER | | Mon Jul 28 1986 21:50 | 12 |
| Re .17:
> Now if point A is able to rotate a right-most tile somewhere into the
> rectangle, we must have 'x1 <= a+y0'.
Okay.
> For point B, we have 'y0 <= x1-b'.
No, it's 'x1 <= b+y0', just like A.
John
|
| 542.21 | More on Lynn's bathroom :-) | CLT::GILBERT | It's a Dusey | Mon Jul 28 1986 23:13 | 48 |
| 542.22 | | CLT::GILBERT | It's a Dusey | Tue Jul 29 1986 19:28 | 60 |
| 542.23 | finding possible centers, given the convex hull | CLT::GILBERT | It's a Dusey | Tue Jul 29 1986 21:48 | 116 |
| 542.24 | | CLT::GILBERT | schmaltzy | Wed Jul 30 1986 23:13 | 68 |
| 542.25 | Centers cannot form an obtuse triangle | CLT::GILBERT | eager like a child | Tue Aug 05 1986 15:13 | 37 |
| 542.26 | Here's the tile-rotation note: | CHOVAX::YOUNG | Where is our Laptop VAXstation? | Sun Oct 21 1990 17:05 | 1 |
| Mentioned at the math dinner...
|