Geomantic Magic Squares
I am fascinated with patterns that have patterns within patterns, wherever I can find them.
Fractals are like that– Simple equations that generate complex patterns, which don’t come out the same with successive runs. Fractals are probably the way that Nature encodes the instructions for making trees. How do the trees know how to do that? Topic for another blog...
The Fibonacci sequence and its related structures are another example of this– Awesome and wonderful, and a model that seems to be used by the ground structure of the Universe, as discussed in Chemistry by Number Theory.
Well, here’s another one--
I have been reading about the ancient practice of Geomancy. Geomancy is a traditional technique that apparently started somewhere in Africa or the Middle East. The idea is to generate a series of four figures, each consisting of four levels of dots. Each level can have one or two dots. Once generated, the four figures are used to answer questions with standard rules. It is sort of like I Ching or Astrology, with the concept that the reader is consulting the “Spiritus Mundi” or planetary consciousness for the answer to the question.
There are sixteen figures in Geomancy. Each one is generated from top to bottom by one of a number of methods, including flipping coins, casting dice or sticks, or whacking sand or wax tablets with a stick. They look like this:
It’s possible to think of each 4-line figure as a binary number. If each line of two dots represents a “zero”, and each line with one dot is a “one,” then the sixteen standard figures represent binary numbers counting from 0 to 15, a total of sixteeen numbers.
I began to wonder if anyone had thought to insert the Geomantic figures into a 4x4 Magic Square. Also, what would be the significance of doing something like that?
One internet search later, I found a blog entry from digitalambler on this topic, who approached it from a completely different point of view. It is well worth reading– Here’s a link:
https://digitalambler.com/2019/02/18/on-geomantic-figure-magic-squares/
Digitalambler notes that the Geomantic figures have 4, 5, 6, 7 or 8 dots. He used the number of dots in each figure to generate a value for the figure and did a thorough evaluation of the creation of magic squares on that basis. I noticed some other features in his evaluation. I put in comments to his post, but since he didn’t post them, I don’t feel bad about putting them on my own blog;
"Fascinating post!
I also notice that the quantities of the geomantic figures from four dots to eight are also described by the fifth level of Pascal’s Triangle, ie.,
1
11
121
1331
14641
ie., one 4, four 5’s, six 6’s, four 7’s, one 8.
Another thought--
There is at least one additional way to generate a geomantic magic square:
Each of the 4 lines of each geomantic figure can be thought of as a ‘one’ or a ‘zero.’ Let the single dot be a ‘1’, and the double dot be a ‘zero.’
In this case, each geomantic figure has a numeric value from 0 to 15– Populus would be ‘0’ and Via would be ’15.’
There is at least one numerical magic square out there with the numbers from 1 to 16, so if you add ‘1’ to each figure’s value, you can plug them into the 1 to 16 magic square.
09 06 03 16
04 15 10 05
14 01 08 11
07 12 13 02 The lines, diagonals and corners add to ‘34.’
You also get a magic square with the numbers from zero to 15, with rows adding to 30– either way the placement of figures is the same:
08 05 02 15
03 14 09 04
13 00 07 10
06 11 12 01
Plugging in the geomantic figures, you get:
Rotating the square 45 degrees into a diamond with Via at the top and Conjunctio at the bottom produces a diamond with mirroring figures;
Albus, Rubeus;
Acquisitio, Carcer, Rubeus;
Laetitia, Cauda Draconis, Caput Draconis, Tristitia;
Fortuna Major, Populus, Fortuna Minor;
Puer, Puella
The symmetries in the diamond-form of this magic square are only apparent when you substitute the Geomantic figures for the numbers. Hmmm...
In this magic square, I have put it into a diamond-configuration. The geomantic figures are vertical, and the 4-digit numbers in each square are the numbers for the magic square in binary. Notice the remarkable binary symmetry in each horizontal row.
Without working too hard at it, I found two more magic circles on the internet that use numbers 1 to 16;
04 14 15 01
09 07 06 12
05 11 10 08
16 02 03 13
08 11 14 01
13 02 07 12
03 16 09 06
10 05 04 15
How many 4x4 Magic Squares could there be?
23 Mar 2019--
PS--
In addition to the above, I found a website that has (probably) published all the possible 4X4 magic squares, in all iterations. There are 880 unique magic squares. Multiply that figure by 8 to account for mirrors, flips and rotations.
NOW--There are 16 geomantic figures, most of which are paired with an inverted figure or a complementary figure.
On the magic squares website, when you select "even/odd" coloration, you find that there are eight patterns, each with a complement (and sometimes a rotation). So it's possible that the Geomantic figures are each represented in a set of magic squares. Hmmm....
When a person casts the 4 figures that generate a geomantic reading, if they are the first 4 characters of a magic square, it could be grounds for additional assessment;
Here's a link:
http://www.cafe64.net/numbers/4x4/4x4.html
The 7040 4x4 magic squares are arranged by the number in the upper left corner.
Click on a number in the 3rd line, "Go to set>>"
On the "Switch View" line, click "Odd/Even."
The odd and even numbers will be differently coloured. You will see eight types of patterns in the even and odd numbers. Some of the patterns, like Carcer, and Fortuna Major/Fortuna Minor, have very similar patterns in the magic squares.
As an example, let's suppose you do a Geomantic reading and cast Tristitia, Laetitia, Carcer, and Via. Reading from top to bottom with two dots=0 and one dot=1 (as previously described), a numeric value for each Geomantic figure can be assigned from 0 to 15. In this case you get 8-1-6-15.
From the table, there are ten magic squares that contain this sequence as the top line. Substitute the numbers of these 10 magic squares with figures, and you might see something interesting.
The four Geomantic figures that make up the vertical axis of the diamond above (ie., 15, 9,0, 6) are the only characters with bilateral symmetry. This is also true of their binary numbers, of course (1111, 1001, 0000, 0110). Looking through the 4x4 magic square tables at cafe64.net above, it seems to be true that any 4x4 magic square with the numbers 15,9,0, and 6 appearing diagonally (in any order) will also exhibit full binary number symmetry in its diamond form.
I was able to identify nine such squares in just the tables with a '15' in the upper left corner--
Their serial numbers on that website were 6782, 6794, 6843, 6821, 6831, 6849, 6852, 6918, 6923.
In the tables with a '9' in the upper left corner, additional examples are #4064 and 4261.
Take home message-- There are a LOT of them!
PPS--
Within the Magic Square universe, there are some Magic Squares that are .... more magical than others. Pan-Magic Squares not only have each row, column and diagonal adding up to the magic sum (in this case 30), but any grouping of four numbers within the square also adds up to 30, as well as the four corners of the square. How many Pan Magic Squares are there?
At this website,
https://www.grogono.com/magic/4x4.php
the authors point out that there are only three types of 4x4 Pan-Magic, magic squares. All the rest of the Pan Magic squares are flips, rotations, or transpositions of them.
00 07 09 14
11 12 02 05
06 01 15 08
13 10 04 03
00 07 10 13
11 12 01 06
05 02 15 08
14 09 04 03
00 07 12 11
13 10 01 06
03 04 15 08
14 09 02 05
Of course, there's more--
In this YouTube video, the speaker describes the symmetry of patterns of lines (or dance steps!) when you connect lines between the numbers in order from 0 to 15.
https://www.youtube.com/watch?v=-Tbd3dzlRnY
Patterns within patterns within patterns. It’s what I like!
What do we do with this? I have no idea, but it’s remarkable and fascinating!
I am fascinated with patterns that have patterns within patterns, wherever I can find them.
Fractals are like that– Simple equations that generate complex patterns, which don’t come out the same with successive runs. Fractals are probably the way that Nature encodes the instructions for making trees. How do the trees know how to do that? Topic for another blog...
The Fibonacci sequence and its related structures are another example of this– Awesome and wonderful, and a model that seems to be used by the ground structure of the Universe, as discussed in Chemistry by Number Theory.
Well, here’s another one--
I have been reading about the ancient practice of Geomancy. Geomancy is a traditional technique that apparently started somewhere in Africa or the Middle East. The idea is to generate a series of four figures, each consisting of four levels of dots. Each level can have one or two dots. Once generated, the four figures are used to answer questions with standard rules. It is sort of like I Ching or Astrology, with the concept that the reader is consulting the “Spiritus Mundi” or planetary consciousness for the answer to the question.
There are sixteen figures in Geomancy. Each one is generated from top to bottom by one of a number of methods, including flipping coins, casting dice or sticks, or whacking sand or wax tablets with a stick. They look like this:
It’s possible to think of each 4-line figure as a binary number. If each line of two dots represents a “zero”, and each line with one dot is a “one,” then the sixteen standard figures represent binary numbers counting from 0 to 15, a total of sixteeen numbers.
I began to wonder if anyone had thought to insert the Geomantic figures into a 4x4 Magic Square. Also, what would be the significance of doing something like that?
One internet search later, I found a blog entry from digitalambler on this topic, who approached it from a completely different point of view. It is well worth reading– Here’s a link:
https://digitalambler.com/2019/02/18/on-geomantic-figure-magic-squares/
Digitalambler notes that the Geomantic figures have 4, 5, 6, 7 or 8 dots. He used the number of dots in each figure to generate a value for the figure and did a thorough evaluation of the creation of magic squares on that basis. I noticed some other features in his evaluation. I put in comments to his post, but since he didn’t post them, I don’t feel bad about putting them on my own blog;
"Fascinating post!
I also notice that the quantities of the geomantic figures from four dots to eight are also described by the fifth level of Pascal’s Triangle, ie.,
1
11
121
1331
14641
ie., one 4, four 5’s, six 6’s, four 7’s, one 8.
Another thought--
There is at least one additional way to generate a geomantic magic square:
Each of the 4 lines of each geomantic figure can be thought of as a ‘one’ or a ‘zero.’ Let the single dot be a ‘1’, and the double dot be a ‘zero.’
In this case, each geomantic figure has a numeric value from 0 to 15– Populus would be ‘0’ and Via would be ’15.’
There is at least one numerical magic square out there with the numbers from 1 to 16, so if you add ‘1’ to each figure’s value, you can plug them into the 1 to 16 magic square.
09 06 03 16
04 15 10 05
14 01 08 11
07 12 13 02 The lines, diagonals and corners add to ‘34.’
You also get a magic square with the numbers from zero to 15, with rows adding to 30– either way the placement of figures is the same:
08 05 02 15
03 14 09 04
13 00 07 10
06 11 12 01
Plugging in the geomantic figures, you get:
Rotating the square 45 degrees into a diamond with Via at the top and Conjunctio at the bottom produces a diamond with mirroring figures;
Albus, Rubeus;
Acquisitio, Carcer, Rubeus;
Laetitia, Cauda Draconis, Caput Draconis, Tristitia;
Fortuna Major, Populus, Fortuna Minor;
Puer, Puella
The symmetries in the diamond-form of this magic square are only apparent when you substitute the Geomantic figures for the numbers. Hmmm...
In this magic square, I have put it into a diamond-configuration. The geomantic figures are vertical, and the 4-digit numbers in each square are the numbers for the magic square in binary. Notice the remarkable binary symmetry in each horizontal row.
Without working too hard at it, I found two more magic circles on the internet that use numbers 1 to 16;
04 14 15 01
09 07 06 12
05 11 10 08
16 02 03 13
08 11 14 01
13 02 07 12
03 16 09 06
10 05 04 15
How many 4x4 Magic Squares could there be?
23 Mar 2019--
PS--
In addition to the above, I found a website that has (probably) published all the possible 4X4 magic squares, in all iterations. There are 880 unique magic squares. Multiply that figure by 8 to account for mirrors, flips and rotations.
NOW--There are 16 geomantic figures, most of which are paired with an inverted figure or a complementary figure.
On the magic squares website, when you select "even/odd" coloration, you find that there are eight patterns, each with a complement (and sometimes a rotation). So it's possible that the Geomantic figures are each represented in a set of magic squares. Hmmm....
When a person casts the 4 figures that generate a geomantic reading, if they are the first 4 characters of a magic square, it could be grounds for additional assessment;
Here's a link:
http://www.cafe64.net/numbers/4x4/4x4.html
The 7040 4x4 magic squares are arranged by the number in the upper left corner.
Click on a number in the 3rd line, "Go to set>>"
On the "Switch View" line, click "Odd/Even."
The odd and even numbers will be differently coloured. You will see eight types of patterns in the even and odd numbers. Some of the patterns, like Carcer, and Fortuna Major/Fortuna Minor, have very similar patterns in the magic squares.
As an example, let's suppose you do a Geomantic reading and cast Tristitia, Laetitia, Carcer, and Via. Reading from top to bottom with two dots=0 and one dot=1 (as previously described), a numeric value for each Geomantic figure can be assigned from 0 to 15. In this case you get 8-1-6-15.
From the table, there are ten magic squares that contain this sequence as the top line. Substitute the numbers of these 10 magic squares with figures, and you might see something interesting.
The four Geomantic figures that make up the vertical axis of the diamond above (ie., 15, 9,0, 6) are the only characters with bilateral symmetry. This is also true of their binary numbers, of course (1111, 1001, 0000, 0110). Looking through the 4x4 magic square tables at cafe64.net above, it seems to be true that any 4x4 magic square with the numbers 15,9,0, and 6 appearing diagonally (in any order) will also exhibit full binary number symmetry in its diamond form.
I was able to identify nine such squares in just the tables with a '15' in the upper left corner--
Their serial numbers on that website were 6782, 6794, 6843, 6821, 6831, 6849, 6852, 6918, 6923.
In the tables with a '9' in the upper left corner, additional examples are #4064 and 4261.
Take home message-- There are a LOT of them!
PPS--
Within the Magic Square universe, there are some Magic Squares that are .... more magical than others. Pan-Magic Squares not only have each row, column and diagonal adding up to the magic sum (in this case 30), but any grouping of four numbers within the square also adds up to 30, as well as the four corners of the square. How many Pan Magic Squares are there?
At this website,
https://www.grogono.com/magic/4x4.php
the authors point out that there are only three types of 4x4 Pan-Magic, magic squares. All the rest of the Pan Magic squares are flips, rotations, or transpositions of them.
00 07 09 14
11 12 02 05
06 01 15 08
13 10 04 03
00 07 10 13
11 12 01 06
05 02 15 08
14 09 04 03
00 07 12 11
13 10 01 06
03 04 15 08
14 09 02 05
Of course, there's more--
In this YouTube video, the speaker describes the symmetry of patterns of lines (or dance steps!) when you connect lines between the numbers in order from 0 to 15.
https://www.youtube.com/watch?v=-Tbd3dzlRnY
Patterns within patterns within patterns. It’s what I like!
What do we do with this? I have no idea, but it’s remarkable and fascinating!
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