Pythagoras and Agrippa

Of right triangles and odd squares

If we look at the center of the odd sized squares in Agrippa’s magic squares (Saturn, Mars, Mercury, Moon), we can spot lengths of the sides of right triangles. By extension, this holds for all odd sized squares built in the same way.

3² + 4² = 5²
5² + 12² = 13²
7² + 24² = 25²
9² + 40² = 41²

27² + 364² = 365²

Of magic triangles

Now if we consider the associated right triangles sides length (Saturn, Mars, Mercury & Venus triangles).

3² = 4 + 5
5² = 12 + 13
7² = 24 + 25
9² = 40 + 41

27² = 364 + 365
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Lovely pattern

T( 6) = 21
T( 66) = 2211
T( 666) = 222111
T( 6666) = 22221111
T( 66666) = 2222211111
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Goetic Squares

Putting together the trigram and the hexagram squares we now have a complete set of 72 Goetic squares. It is left as an exercise to map each one to its corresponding demon.

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Hexagram or I Ching Squares

After the trigram squares, looking for a further transformation of the LuoShu, we can relax the constraint on the sum of the diagonals. This gives us 64 extra squares that could be mapped on the 64 hexagrams of the I Ching.

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Taijitu Transformations

Taijitu and LuoShu are symbols of balance and harmony. We can apply the same three basic transformations to the Taijitu as we did to the LuoShu: rotation, horizontal mirroring and vertical mirroring.

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Trigrams on a Square!

Let’s place the eight trigrams on a LuoShu square. First let’s associate the trigrams to a value according to their binary representation equivalent.

We need nine numbers and zero was not much in use when the LuoShu was designed. So let’s start counting from one and insert a taijitu in the middle.

The trigrams are now split in two groups that are negative image of each other. The negative image pairing is reinforced by the value associated to each trigram. The sum of each pair is ten.

Once placed according to a LuoShu pattern, the result is a LuoShu Bagua Square.

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Trigram Squares

The Luoshu layout follows the  traditional convention of chinese maps, where North is at the bottom and South at the top, so number one is drawn at the bottom center and number nine at the top center. Still, this is not the only possible layout for a 3×3 magic square and we can take avantage of this to make a link to another chinese concept, the trigrams.

We can use the trigrams as binary encoding of the three transformations: rotation, horizontal mirroring and vertical mirroring. Thus assigning each one of the eight trigrams to one of the eight 3×3 magic squares.

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Taijitu as a mnemonic for LuoShu

While the LuoShu (3×3 magic square) is simple to memorize by itself, one visual trick to remember the layout of the nine numbers is the Taijitu (yin-yang symbol). Write down the numbers from one to nine following the swirling pattern then exchange two and eight.

One to four belong to the shadow, while six to nine are on the sunny side.

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Asinus asinum fricat

Back Rub
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A riddle: I cast dices, pick dominoes, play roulette, draw tarot cards and mahjong tiles but always find the same numbers…

… Triangular numbers.

T( 3) = 6
T( 7) = 28
T( 8) = 36
T( 12) = 78
T( 16) = 136

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