# meditationatae

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## Data on possible/probable 15-vertex, 37-edge unit distance graph

Note: copied from the output of a computer program.

===

adjacency matrix:
0 0 0 0 0 1 1 1 0 0 0 0 1 0 0
0 0 1 0 1 1 0 0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 1 0 0 1 0 0 0 1
0 0 0 0 0 1 0 1 0 1 1 1 0 0 0
0 1 0 0 0 0 1 1 0 1 0 0 0 1 0
1 1 0 1 0 0 0 0 0 0 0 1 1 0 0
1 0 0 0 1 0 0 1 1 0 0 1 0 0 1
1 0 1 1 1 0 1 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0 0 0 1 1 1 1 0
0 0 0 1 1 0 0 0 0 0 0 1 0 1 0
0 0 1 1 0 0 0 1 1 0 0 0 1 1 0
0 0 0 1 0 1 1 0 1 1 0 0 0 0 0
1 0 0 0 0 1 0 0 1 0 1 0 0 0 0
0 1 0 0 1 0 0 0 1 1 1 0 0 0 0
0 0 1 0 0 0 1 0 0 0 0 0 0 0 0

post adjacency matrix:
0 0 0 0 0 1 1 1 0 0 0 0 1 0 0
0 0 1 0 1 1 0 0 0 0 0 0 0 1 1
0 1 0 0 0 0 0 1 0 0 1 0 0 0 1
0 0 0 0 0 1 0 1 0 1 1 1 0 0 0
0 1 0 0 0 0 1 1 0 1 0 0 0 1 0
1 1 0 1 0 0 0 0 0 0 0 1 1 0 0
1 0 0 0 1 0 0 1 1 0 0 1 0 0 1
1 0 1 1 1 0 1 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0 0 0 1 1 1 1 1
0 0 0 1 1 0 0 0 0 0 0 1 0 1 0
0 0 1 1 0 0 0 1 1 0 0 0 1 1 0
0 0 0 1 0 1 1 0 1 1 0 0 0 0 0
1 0 0 0 0 1 0 0 1 0 1 0 0 0 0
0 1 0 0 1 0 0 0 1 1 1 0 0 0 0
0 1 1 0 0 0 1 0 1 0 0 0 0 0 0

total_finds = 43
Tests count = 15 has_converged_step = 2616 max_has_converged_step = 2616
rec_diff1 = 39.857967
rec_diff25 = 5.935028
rec_diff50 = 2.601235
rec_diff100 = 0.381244
rec_diff200 = 0.069902
rec_diff400 = 0.005289
rec_diff800 = 0.0000311393226121
rec_diff1200 = 0.0000001835105382
probable_unit_lengths = 37
graph is 3-colorable:no.
count_4_colorings = 364 num_4_colorings_mod_24 = 0
Matrix of distances vertex to vertex:
A B C D E F G H I J K L M N O
A 0.00 1.57 0.58 1.95 1.73 1.00 1.00 1.00 1.95 2.73 1.46 1.91 1.00 2.43 1.46
B 1.57 0.00 1.00 1.57 1.00 1.00 0.58 1.46 0.46 1.73 0.58 0.74 0.74 1.00 1.00
C 0.58 1.00 0.00 1.73 1.37 0.74 0.46 1.00 1.37 2.35 1.00 1.44 0.46 1.91 1.00
D 1.95 1.57 1.73 0.00 0.58 1.00 1.46 1.00 1.95 1.00 1.00 1.00 1.91 1.46 2.43
E 1.73 1.00 1.37 0.58 0.00 0.74 1.00 1.00 1.37 1.00 0.46 0.46 1.44 1.00 1.91
F 1.00 1.00 0.74 1.00 0.74 0.00 0.58 0.46 1.46 1.73 0.58 1.00 1.00 1.57 1.57
G 1.00 0.58 0.46 1.46 1.00 0.58 0.00 1.00 1.00 1.95 0.58 1.00 0.46 1.46 1.00
H 1.00 1.46 1.00 1.00 1.00 0.46 1.00 0.00 1.91 1.91 1.00 1.37 1.37 1.95 1.95
I 1.95 0.46 1.37 1.95 1.37 1.46 1.00 1.91 0.00 1.91 1.00 1.00 1.00 1.00 1.00
J 2.73 1.73 2.35 1.00 1.00 1.73 1.95 1.91 1.91 0.00 1.37 1.00 2.35 1.00 2.73
K 1.46 0.58 1.00 1.00 0.46 0.58 0.58 1.00 1.00 1.37 0.00 0.46 1.00 1.00 1.46
L 1.91 0.74 1.44 1.00 0.46 1.00 1.00 1.37 1.00 1.00 0.46 0.00 1.37 0.58 1.73
M 1.00 0.74 0.46 1.91 1.44 1.00 0.46 1.37 1.00 2.35 1.00 1.37 0.00 1.73 0.58
N 2.43 1.00 1.91 1.46 1.00 1.57 1.46 1.95 1.00 1.00 1.00 0.58 1.73 0.00 1.95
O 1.46 1.00 1.00 2.43 1.91 1.57 1.00 1.95 1.00 2.73 1.46 1.73 0.58 1.95 0.00

TBC …

===

Added:  Mon Mar 20 04:02:36 EDT 2017

A reminder:

Thomas Sauvaget who has a blog named

episodic thoughts

found a 16-vertex graph consisting approximately of two

Moser spindles, plus two more points in a special

configuration:

https://thomas1111.wordpress.com/2015/01/03/on-a-certain-4-chromatic-planar-graph/

He writes in part:

As it happens, if we enforce unit distances at 6~14 and 10~14 then the abscissa of vertex #14 is not $0.5$ (which would make things work), but instead

$-\frac{895702085}{177666094271969928} \sqrt{514527538436665}+ \frac{415413875483399597}{676176985277553384}$ $\approx 0.49999956608$  (and same problem with vertex #15).  ”

===

So, I’m thinking that I should not be assuming that vertices in an embedded graph are a unit apart, just because they are 1.0 units apart to within 0.0000001  …

Perhaps I can use the MPFR library, or PARI/gp to

calculate distances between points that appear to be 1.0 units apart, to 50 or more decimals or significant figures …

Written by meditationatae

March 20, 2017 at 8:00 am

Posted in History

## Image of a 15-vertex 35-edge possible unit distance graph

Written by meditationatae

March 18, 2017 at 10:54 pm

Posted in History

## Using the ldd command with the verbose option

With respect to the readline library, some versions have various symbols (UP, etc.) defined, and some do not.

For PARI/gp on my system, it only works when these various symbols are defined.

PARI/gp question: how does one get a history of the commands input at the terminal ?

We are in the directory:  /usr/local/lib .

ls -l gives output:

-rw-r–r–. 1 root root 1291398 Feb 22 12:51 libgmp.a
-rwxr-xr-x. 1 root root 913 Feb 22 12:51 libgmp.la
lrwxrwxrwx. 1 root root 16 Feb 22 12:51 libgmp.so -> libgmp.so.10.3.2
lrwxrwxrwx. 1 root root 16 Feb 22 12:51 libgmp.so.10 -> libgmp.so.10.3.2
-rwxr-xr-x. 1 root root 523381 Feb 22 12:51 libgmp.so.10.3.2
-rw-r–r–. 1 root root 162874 Feb 23 02:55 libhistory.a
-rw-r–r–. 1 root root 171982 Feb 22 14:11 libhistory.old
lrwxrwxrwx. 1 root root 15 Feb 23 02:55 libhistory.so -> libhistory.so.6
lrwxrwxrwx. 1 root root 17 Feb 23 02:55 libhistory.so.6 -> libhistory.so.6.3
-r-xr-xr-x. 1 root root 99093 Feb 23 02:55 libhistory.so.6.3
lrwxrwxrwx. 1 root root 21 Feb 22 14:11 libhistory.so.7 -> libhistory.so.7.0.old
-r-xr-xr-x. 1 root root 106376 Feb 22 14:11 libhistory.so.7.0
-r-xr-xr-x. 1 root root 106376 Feb 22 12:35 libhistory.so.7.0.old
-rwxr-xr-x. 1 root root 7060519 Feb 23 11:55 libpari-gmp.so.2.9.1
lrwxrwxrwx. 1 root root 20 Feb 23 11:55 libpari-gmp.so.5 -> libpari-gmp.so.2.9.1
lrwxrwxrwx. 1 root root 20 Feb 23 11:55 libpari.so -> libpari-gmp.so.2.9.1
-rw-r–r–. 1 root root 1199592 Feb 23 02:55 libreadline.a
-rw-r–r–. 1 root root 1239926 Feb 22 14:11 libreadline.old
lrwxrwxrwx. 1 root root 16 Feb 23 02:55 libreadline.so -> libreadline.so.6
lrwxrwxrwx. 1 root root 25 Mar 10 08:32 libreadline.so.6 -> /lib64/libreadline.so.6.0
-r-xr-xr-x. 1 root root 680148 Feb 23 02:55 libreadline.so.6.3
lrwxrwxrwx. 1 root root 22 Feb 22 14:11 libreadline.so.7 -> libreadline.so.7.0.old
-r-xr-xr-x. 1 root root 702910 Feb 22 14:11 libreadline.so.7.0
-r-xr-xr-x. 1 root root 702910 Feb 22 12:35 libreadline.so.7.0.old
drwxr-xr-x. 2 root root 4096 Feb 23 11:55 pari

I single out the libreadline text:

lrwxrwxrwx. 1 root root 16 Feb 23 02:55 libreadline.so -> libreadline.so.6
lrwxrwxrwx. 1 root root 25 Mar 10 08:32 libreadline.so.6 -> /lib64/libreadline.so.6.0

The ‘l’ signifies a symbolic link, so that

libreadline.so is an alias name for libreadline.so.6, and

libreadline.so.6 is an alias name for /lib64/libreadline.so.6.0 , and

/lib64/libreadline.so.6.0 is a normal file:

$ls -l /lib64/libreadline.so.6.0 -rwxr-xr-x. 1 root root 272008 Jun 22 2012 /lib64/libreadline.so.6.0 (272 kbytes ). Now, I try:$ ldd -v libreadline.so  and I get this:

linux-vdso.so.1 => (0x00007ffd3c5fc000)
libtinfo.so.5 => /lib64/libtinfo.so.5 (0x0000003b0fa00000)
libc.so.6 => /lib64/libc.so.6 (0x0000003afce00000)
/lib64/ld-linux-x86-64.so.2 (0x0000003afc600000)

Version information:
./libreadline.so:
libc.so.6 (GLIBC_2.4) => /lib64/libc.so.6
libc.so.6 (GLIBC_2.3) => /lib64/libc.so.6
libc.so.6 (GLIBC_2.3.4) => /lib64/libc.so.6
libc.so.6 (GLIBC_2.11) => /lib64/libc.so.6
libc.so.6 (GLIBC_2.2.5) => /lib64/libc.so.6
/lib64/libtinfo.so.5:
libc.so.6 (GLIBC_2.4) => /lib64/libc.so.6
libc.so.6 (GLIBC_2.3) => /lib64/libc.so.6
libc.so.6 (GLIBC_2.3.4) => /lib64/libc.so.6
libc.so.6 (GLIBC_2.2.5) => /lib64/libc.so.6
/lib64/libc.so.6:
ld-linux-x86-64.so.2 (GLIBC_PRIVATE) => /lib64/ld-linux-x86-64.so.2
ld-linux-x86-64.so.2 (GLIBC_2.3) => /lib64/ld-linux-x86-64.so.2

Note: there are no missing symbols.

What about libreadline.so.6.3 ? It looks just as good…

$ldd -v libreadline.so.6.3 gives output: linux-vdso.so.1 => (0x00007ffdd6951000) libc.so.6 => /lib64/libc.so.6 (0x00007f815ed7e000) /lib64/ld-linux-x86-64.so.2 (0x0000003afc600000) Version information: ./libreadline.so.6.3: libc.so.6 (GLIBC_2.3) => /lib64/libc.so.6 libc.so.6 (GLIBC_2.2.5) => /lib64/libc.so.6 /lib64/libc.so.6: ld-linux-x86-64.so.2 (GLIBC_PRIVATE) => /lib64/ld-linux-x86-64.so.2 ld-linux-x86-64.so.2 (GLIBC_2.3) => /lib64/ld-linux-x86-64.so.2 === It looks good too, although /lib64/libtinfo.so.5 present in first is what defines UP etc. === Another try, this time with: libreadline.so.7.0$ ldd -v libreadline.so.7.0
linux-vdso.so.1 => (0x00007fff4e3b2000)
libc.so.6 => /lib64/libc.so.6 (0x00007f301dec9000)
/lib64/ld-linux-x86-64.so.2 (0x0000003afc600000)

Version information:
./libreadline.so.7.0:
libc.so.6 (GLIBC_2.3) => /lib64/libc.so.6
libc.so.6 (GLIBC_2.2.5) => /lib64/libc.so.6
/lib64/libc.so.6:
ld-linux-x86-64.so.2 (GLIBC_PRIVATE) => /lib64/ld-linux-x86-64.so.2
ld-linux-x86-64.so.2 (GLIBC_2.3) => /lib64/ld-linux-x86-64.so.2

===

this one also doesn’t have

/lib64/libtinfo.so.5.

===

linux-vdso.so.1 => (0x00007ffe2756a000)
libc.so.6 => /lib64/libc.so.6 (0x0000003afce00000)
/lib64/ld-linux-x86-64.so.2 (0x0000003afc600000)

Version information:
/lib64/libtinfo.so.5:
libc.so.6 (GLIBC_2.4) => /lib64/libc.so.6
libc.so.6 (GLIBC_2.3) => /lib64/libc.so.6
libc.so.6 (GLIBC_2.3.4) => /lib64/libc.so.6
libc.so.6 (GLIBC_2.2.5) => /lib64/libc.so.6
/lib64/libc.so.6:
ld-linux-x86-64.so.2 (GLIBC_PRIVATE) => /lib64/ld-linux-x86-64.so.2
ld-linux-x86-64.so.2 (GLIBC_2.3) => /lib64/ld-linux-x86-64.so.2

===

Written by meditationatae

March 17, 2017 at 6:12 pm

Posted in History

## Re: “Best candidate” graph, 14 vertices, udg

The post from a few weeks ago shows a 14-vertex 33-edge probable unit distance graph with 324 4-colorings.

The URL of that blog post =

https://meditationatae.wordpress.com/2017/02/20/best-candidate-graph-14-vertices-udg/

Written by meditationatae

March 14, 2017 at 2:46 pm

Posted in History

## Figure of a 13-vertex 28-edge graph

It’s a probable unit distance graph.

It has 356 4-colorings, and is not 3-colorable.

Its adjacency matrix is:

0 0 0 0 0 0 0 0 1 0 0 1 1
0 0 0 0 0 1 0 1 1 1 0 0 1
0 0 0 1 0 1 0 0 1 0 1 0 0
0 0 1 0 0 0 1 1 1 0 0 0 0
0 0 0 0 0 0 1 0 0 0 1 1 0
0 1 1 0 0 0 0 1 0 0 1 1 0
0 0 0 1 1 0 0 1 0 0 1 0 1
0 1 0 1 0 1 1 0 0 0 0 1 0
1 1 1 1 0 0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0 0 0 0 0 1
0 0 1 0 1 1 1 0 0 0 0 0 1
1 0 0 0 1 1 0 1 0 0 0 0 0
1 1 0 0 0 0 1 0 1 1 1 0 0

It also contains as a subgraph the Moser spindle; one of the vertices of the Moser spindle subgraph is the degree 4 (in the Moser spindle) vertex that is uppermost in the figure. Five of the seven vertices in the Moser spindle together with 5 edges amongst these 5 vertices make up a pentagon with bilateral symmetry; by staring enough at the figure, it can be found by me without too much trouble:

Written by meditationatae

March 10, 2017 at 2:58 pm

Posted in History

## My first test-drive using exiftool on Windows

I had read that EXIF metadata in files that are images, whether .jpg or other file formats, have embbedded data, sometimes inluding geotags (coordinates).

Below is the copy/paste of my first test running exiftool.exe on Windows 10:

C:\Users\abcde\Downloads> C:\Users\abcde\Downloads\exiftool -All Louvre_Museum_Wikimedia_Commons.jpg

ExifTool Version Number : 10.45
File Name : Louvre_Museum_Wikimedia_Commons.jpg
Directory : .
File Size : 53 kB
File Modification Date/Time : 2017:03:08 07:18:37-05:00
File Access Date/Time : 2017:03:08 07:18:37-05:00
File Creation Date/Time : 2017:03:08 07:18:36-05:00
File Permissions : rw-rw-rw-
File Type : JPEG
File Type Extension : jpg
MIME Type : image/jpeg
JFIF Version : 1.01
Resolution Unit : inches
X Resolution : 240
Y Resolution : 240
Comment : File source: https://commons.wikimedia.org/wiki/File:Louvre_Museum_Wikimedia_Commons.jpg
Profile CMM Type : lcms
Profile Version : 2.1.0
Profile Class : Display Device Profile
Color Space Data : RGB
Profile Connection Space : XYZ
Profile Date Time : 2012:01:25 03:41:57
Profile File Signature : acsp
Primary Platform : Apple Computer Inc.
CMM Flags : Not Embedded, Independent
Device Manufacturer :
Device Model :
Device Attributes : Reflective, Glossy, Positive, Color
Rendering Intent : Perceptual
Connection Space Illuminant : 0.9642 1 0.82491
Profile Creator : lcms
Profile ID : 0
Profile Description : c2
Profile Copyright : FB
Media White Point : 0.9642 1 0.82491
Media Black Point : 0.01205 0.0125 0.01031
Red Matrix Column : 0.43607 0.22249 0.01392
Green Matrix Column : 0.38515 0.71687 0.09708
Blue Matrix Column : 0.14307 0.06061 0.7141
Red Tone Reproduction Curve : (Binary data 64 bytes, use -b option to extract)
Green Tone Reproduction Curve : (Binary data 64 bytes, use -b option to extract)
Blue Tone Reproduction Curve : (Binary data 64 bytes, use -b option to extract)
Image Width : 800
Image Height : 336
Encoding Process : Baseline DCT, Huffman coding
Bits Per Sample : 8
Color Components : 3
Y Cb Cr Sub Sampling : YCbCr4:2:0 (2 2)
Image Size : 800×336
Megapixels : 0.269

Written by meditationatae

March 8, 2017 at 12:41 pm

Posted in History

## A 12-vertex, 26-edge, probable unit distance graph

The 12-vertex, 26-edge graph shown above is not 3-colorable, and is 4-colorable in 178 ways, up to permutation of the four colors. Numerical evidence suggests that it is a unit distance graph, i.e. embeddable in the plane with all edges of unit length.

My unit distance graph solver found no 12-vertex, non 3-colorable, 4-colorable udg with less than 178 different colorings, counting permutations of the four colors as equivalent. The search was not exahaustive and lasted a day.

Below, I copy the adjacency matrix for the graph above. The rows and columns are not labeled, but it is easy to guess: A for row 1, B for row 2, C for row 3, …. L for row 12; similarly for the labeling of the columns.

I don’t know how to do a fixed-width, or monospace font, in WordPress, so I omit the labels to avoid misalignment.

Adjacency matrix, 12 by 12:

0 1 0 0 0 1 1 1 1 0 0 0
1 0 0 0 0 0 0 0 1 0 1 1
0 0 0 1 0 1 1 0 0 1 0 1
0 0 1 0 1 1 0 0 1 0 0 0
0 0 0 1 0 0 0 1 1 1 1 0
1 0 1 1 0 0 0 1 0 0 0 0
1 0 1 0 0 0 0 0 1 1 0 0
1 0 0 0 1 1 0 0 0 1 1 0
1 1 0 1 1 0 1 0 0 0 0 0
0 0 1 0 1 0 1 1 0 0 0 1
0 1 0 0 1 0 0 1 0 0 0 0
0 1 1 0 0 0 0 0 0 1 0 0

Written by meditationatae

March 4, 2017 at 8:28 pm

Posted in History