I Ching Algorithm: The hexagram space theorem


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Data Mining
There is another type of hexagram that we have not analyzed yet, which is the support hexagram. It is a double hexagram that carries information that supports the master hexagram. Currently, the support hexagram has not been applied in the I Ching Algorithm but will be added if needed.
The support hexagram extracts data from the master hexagram to get supporting knowledge. This is a data mining procedure. We return to the example of the master hexagram HỎA THIÊN ĐẠI HỮU

People omitted dime 1 (ie lower dime) and dime 6 (ie upper dime). These two dimes are at the border, bearing the unique character of the hexagram. The remaining four dimes are the 2nd, 3rd, 4th, and 5th dimes, keeping the basic properties of the original hexagram used to generate the support hexagram as follows:
Take the dime 2, dime 3 and dime 4 of the original hexagram as the lower hexagram of the support hexagram. Take the dime 3, dime 4 and dime 5 of the original hexagram as the upper hexagram of the support hexagram. The result is the hexagram TRẠCH THIÊN QUẢI

The support hexagram ĐOÀI over CÀN are all metal, the ratio is a draw, the hexagram is good. This hexagram has 5 energetic yang dimes, only one yin that is about to die is yang to be prosperous, yin to be decline. The hexagram TRẠCH THIÊN QUẢI is resoluteness and effectively supports the master hexagram. The master hexagram is good, even better.
We have noticed that dimes 3 and 4 are repeated, which emphasizes core information in the center and omits scattered information at the borders.
This reminds us of the convolution technique that uses a window striding over the data space

Both methods represent key feature extraction from the source. However, the convolution technique mines on raw data source and  the support hexagram mines on the dimes with dense knowledge. The convolution technique takes the main feature in terms of data distribution and preserves the overall picture. The support hexagram carries philosophical colors and do not preserve information. Because the dimes already contain dense knowledge, we cannot calculate on them. The lack of upper dime and lower dime of the original hexagram lost information and in fact became a different hexagram from the original hexagram.
However, the support hexagram retains and strengthens the middle dimes that are at their most energetic, it has the most common characteristics of the original hexagram, thus becoming a teammate that can help the master.
While the convolution technique takes a representative of average of the data, the goal of the hexagram is must not be the same as the original hexagram. It cannot be substituted for the original hexagram, but only a support team hexagram.
In short, convolution is a transformation that is data-equivalent but not knowledge-equivalent, it creates knowledge from data. Meanwhile, the establishment of the support hexagram is a special I Ching transformation in the transformation of things in the hexagram space consisting only of knowledge. Transformations in the data space including also data in the form of elementary knowledge cannot "talk" to hexagrams. In the hexagram space, there is only "talk" about hexagram words, dime words and I Ching transformation, is condensed knowledges that has been summed up into truth.
(the hexagram words, the dime words are conclusions about the meaning of hexagrams and dimes, the I Ching transformation is to change the dimes to become another hexagram, taking on a different shape in the process of growing things)
For example, the binary sequence 111 looks like CÀN hexagram, but not a hexagram. It is merely data, its decimal value is 7. The CÀN hexagram is also denoted the same, but its hexagram number is 1. This is the number formed when considering spatial balance and balance yin and yang in the process human interaction with the universe.
Primary knowledge still has to go through one more step: embedding in hexagram space through signals from a particular event.

The hexagram space theorem
Any binary sequence can be embedded in the hexagram space.
Proof: Convolution significantly reduces the size of the data space. When the window pane strides over a picture (2D convolution), the data obtained through the filter is only a few pixels and therefore the resulting picture is very small compared to its original size. If we only look at the window locally we see that a lot of information seems to be lost. But the overall picture remains its character. That's because raw data has been mined to become quintessential knowledge. Convolution is an equivalent transformation.
We're back to the binary sequence
1000101110110101000111
The binary sequence is 22 bits long. We use convolution to generate 6 bits as raw material for 2 single hexagrams. Choose stride = 3, apply the formula
len_w = len_in - (len_out - 1) * stride
We have len_in = 22; len_out = 6
len_w = 22 – (6 – 1) * 3 = 22 – 5 * 3 = 22 – 15 = 7
So the window size is 7.
The filter has a mean value type, which is a vector [1/7, 1/7, 1/7, 1/7, 1/7, 1/7, 1/7]
‘100’010’1,11’0,11’0,10’1,000,111
The six binary segments in the extracted window are
1000101 0101110 1110110 0110101 0101000 1000111
The output is a 6-element vector [3/7, 4/7, 5/7, 4/7, 2/7, 4/7]
Applying the Theorem of problem space representation, we get a 6-bit binary sequence
011101
Note that the above binary sequence is not a double hexagram, it is just elementary knowledge whose size of a double hexagram or two single hexagrams.
The 6-bit binary sequence carries the signal into hexagram space, we halve it into two 3-bit subsequences
011 101
The sequence 011 has a decimal value of 3 corresponding to the hexagram number of the hexagram LY, it is embedded in the hexagram space and we get the upper hexagram LY. The sequence 101 has a decimal value of 5, we have the lower hexagram TỐN. The double hexagram is HỎA PHONG ĐỈNH (the cauldron for cooking, the wind blowing for the fire to flare up). This hexagram is a good hexagram, solid on the cauldron, can cook living things into dead things, turn hard things into soft things, is the hexagram of power

The reason we compress the data to 3-bit form to have the size of a single hexagram, this is an operation to normalize the data size and make full use of the information from the data of the natural event instead of division by 8 to get remainder. The division by 8 to get remainder actually only uses the last 3 bits of a large binary sequence, which is forced because there will be infinitely many large values that share the last 3 bits all giving the same hexagram. Indeed, the ancients did not know the full convolution technique. The introduction of convolution into the I Ching should be seen as a radical improvement in divination.

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