Taijitu Variation

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A calendar: The 120 days of Sodom

You may have read or decided not to read the 120 days of Sodom by Sade. But have you ever wondered why he settled for that number of 120? Expanding on the path laid out by Boccacio and Chaucer, he sets four wealthy and powerful men to gather in a remote schloss to listen to five stories a day during one hundred and twenty days, thus writing down a catalogue of six hundred ‘passions’.

The stories telling part starts on the first day of November to end on the last day of February, so 30+31+31+28 gives 120, voila! As the narration continues until the first day of Spring when they can leave the snowed in schloss we know the story does not end on a leap year, even if “the 121 days of Sodom” would have been cool in a nerdy way.

As the novel goes those four wealthy and powerful men actually arrive at the schloss in the evening of October 29 and would not leave sooner than March 21. This tallies to 3+120+21 or 144 days. Coincidence? Well Sade never refers to one hundred and forty four as a number of days, but he tells us this is the count of female candidates gathered during the preparation to the reunion. Actually it seems so obsessed by this number that he messed up the explanation of this selection, giving 144 which is 16×9 as the total of gathering nine candidates for each of the sixteen french province plus Paris. Sadly he misses the correct count of one hundred and fifty three which is the sum from one to seventeen.

Life at the schloss follows a weekly routine, with a feast every saturday. Starting with the first “day” on sunday November First, the story proceeds during seventeen weeks plus one day, as the last feast is postponed to the following sunday for a total of one hundred and twenty days. As we already noticed that the story does not end on a leap year, it is possible to look further if there is some clue that would help identify the likely date of those fictional events.

In the introduction, Sade places the story at the end of Louis XIV reign ( who died in Septembre 1715) and close to the time when the Chambre de Justice prosecuted abusers of the royal finances (March 1716 to March 1717). Only two years have November first falling on Sunday while not being followed by a leap year: 1705 and 1716.

This means that the main events of the story take place from Sunday November 1 1716 to Sunday February 28 1717.

One last thing: this November first 2020 is a Sunday and 2021 is not a leap year.

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Daoist Magic Talisman

Daoist magic talisman to ward off the evil influences of the “Five Poisonous Animals”.

The luoshu numbers are in dotted form, overlayed on a bagua pattern.

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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

Of squares summed

Then looking at the center square of each odd sized squares, we can see that they can be written as sum of two squares.

5 = 1 ² + 2 ²
13 = 2 ²+ 3 ²
25 = 3 ²+ 4 ²
41 = 4 ²+ 5 ²

365 = 13 ² + 14 ²

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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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