Torsten Sillke, 17. Apr. 96
For a subset of R^n call the multiset of pairwise distances its spectrum.
Two sets are called isospectral, if they have the same spectra.
Problem: Find non-congruent isospectral sets for dimension 1 and 2.
----------------------
2-Dimensional Example:
----------------------
The smallest example of (non-congruent) isospectral sets. (2. Apr. 96)
Take 4 points of the vertices of a regular 8-gon.
X -- X . -- . Both sets are symmetric
/ \ / \ They are complements
. . X X
| | | |
. X . X
\ / \ /
. -- X . -- X
Distance Spectra: 1 2 3 4 (Distancesmassure is arc-lenght)
Count: 2 1 2 1
Partial Classification of 4-point isospectral sets.
---------------------------------------------------
1) 4-Subsets of the circle with circumfence 2a+2b: (5. Apr. 96)
Set 1: 0 a 2a+b 2a+2b (Arclength distance)
a a+b b
2a+b a+2b
2a+2b
Set 2: 0 a 2a+2b 4a+3b
a a+2b 2a+b
2a+2b a+b
b
A 5th point may be added at the center. (eucledian distance)
2) symmetric 4-Subsets (12. Apr. 1996)
Set 1: (a,0), (-a,0), (b,b), (-b,b) (5th point optional (0,0))
Set 2: (b,0), (-b,0), (0,a+b), (0,-a+b) (5th point optional (0,b))
Special cases of this family are:
x x x x x
. x . x
x
found by ksbrown@ksbrown.seanet.com (Kevin Brown)
x x x x
x . . x
x x
found by Life1ine@aol.com (Bob Wainwright)
At '.' a fifth point may be added.
My 8-gon example is a special case to 1) and 2).
3) maximal families of 4-Subsets (17. Apr. 96)
Set 1: (-c,0), (c,0), (-d*b, d*a), (a,b) (5th point optional (0,0))
Set 2: (-c,0), (c,0), (-d*b, d*a), (-a,-b) (5th point optional (0,0))
A coordinate free describtion is as follows.
Let X, Y, Z be nonzero points in R^2 with Y*Z = 0. Then the multisets
{ -X, X, Y, Z } and { -X, X, Y, -Z } are isospectral.
This family has maximal degree of freedom.
Only two distances exchange.
Special cases of this family are:
x
x x x x
. .
x x
x
found by Life1ine@aol.com (Bob Wainwright)
At '.' a fifth point may be added. (found by me)
Open Problems:
Classify all 4-point isospectral sets.
All 4-point isospectral sets found so far have the property,
that a 5th point can be added. Does this hold in all cases?
Tripple isospectral sets:
-------------------------
Kevin Brown found three sets which are isospectral.
* * * * * * * * * *
* * * *
* * * * *
* * * * *
----------------------
1-Dimensional Example:
----------------------
A 6-point isospectral example:
X1 = { 0, 1, 2, 6, 8, 11 }
X2 = { 0, 1, 6, 7, 9, 11 }
Distance Spectra: 1 2 3 4 5 6 7 8 9 10 11
Count: 2 2 1 1 2 2 1 1 1 1 1
The smallest diameter of a isospectral set in Z is 11.
6-point isospectral families: (9. Apr. 96)
X1 = { 0, a, a+b, 4a+2b, 6a+2b, 8a+3b }
X2 = { 0, a, 5a+b, 5a+2b, 7a+2b, 8a+3b }
Distances:
3b . . . . . . . 1 1
2b . . . 1 1 1 1 1 .
1b 1 1 1 1 1 1 . . .
0b . 1 1 . . . . . .
0a 1a 2a 3a 4a 5a 6a 7a 8a
Y1 = { 0, a, a+b, 4a+2b, 6a+4b, 8a+5b }
Y2 = { 0, a, 3a+b, 3a+2b, 7a+4b, 8a+5b }
Distances:
5b . . . . . . . 1 1
4b . . . . . 1 1 1 .
3b . . . . 1 1 . . .
2b . . 1 1 1 . . . .
1b 1 1 1 1 . . . . .
0b . 1 . . . . . . .
0a 1a 2a 3a 4a 5a 6a 7a 8a
Theorem: There are no 4-point isospectral sets. (5. Apr. 96)
Open Problems:
--------------
Are six points minimal? (I tried 5 points upto diameter 31.)
Classify all 6-point isospectral sets.
(This can be done by calculating several LP-problems.)
Are there tripples of isospectral sets?
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Article 13147 of sci.math:
From: edp@math.zko.dec.com (Eric Postpischil)
Subject: Re: Scientists as Programmers (was Re: Small Language Wanted)
Organization: Digital Equipment Corporation
Date: Tue, 1 Sep 1992 17:36:36 GMT
[...]
As it happens, I am currently investigating homometric rulers that
measure distinct distances; these are part of a class of problems with
applications in crystallography, coherence theory, signal processing,
antenna array design, wavefront estimation in speckle imaging, electron
microscopy, inverse scattery, and image recovery from speckle
interferometry data in astronmy. The problem I am examining is this:
Given a ruler with m marks at various location which can measure the
m*(m-1)/2 distances between each pair of marks, is there another ruler
that measures the same distances but is not produced by reflection or
translation of the first? It is known there are no such rulers for 5 or
fewer marks, but there is one family of such ruler pairs with 6 marks.
Bloom and Golomb have conducted numeric searches for pairs with 7, 8,
and 9 marks with lengths up to 33, 43, and 49, respectively. But a
colleague and I, both software engineers, have used non-numeric
algorithms to search for symbolic solutions, and have proven there are
no such rulers with 7, 8, 9, or 10 marks of ANY length. (Strictly, the
algorithms we use are numeric, in the coefficients of the algebraic
expressions we manipulate, but no numbers for the mark locations appear
in the program at any time; they are always symbolic.) The numeric
programs could have been speed up a trillion times and they never would
have produced our results.
-- edp (Eric Postpischil)
"Always mount a scratch monkey."
edp@alien.enet.dec.com