In Africa I spent many days on a truck, where it was too bumpy to read and too noisy to listen to lectures.  Perfect time to bite the bullet on the Tamariz stack.  Initially I approached Mnemonica using mnemonics, by generating adjective:noun pairing (with Mnemonics Minor) for each index:value pairing. (For example, given f stands in for 8 and s stands in for 5, foxy spy would correspond to position 8, value 5.)  This turned out to be a losing proposition, for two reasons.  The first is that I ended up with solid memory of the mnemonic phrase, but not the actual pairing; this meant that I needed to decode the actual pairing every time, and it’s too slow for stack work.

The second problem with this method is that I needed to separately remember the suit, for which the mnemonic phrases gave no clues to.  I had a hunch that there would be order within the suits, since (1) the operations used to generate Mnemonica — faro, reversals — were high-symmetry operations, and (2) it is known that complete suits could be dealt, given minor adjustments.  We’ll look at this quasi-order within Mnemonica here.

The first thing I did was to lay out the order in a wheel; afterall, Mnemonica is cyclical with a haphazard beginning (why does it end at 9D?)  I additionally marked the portion which had an extra partial faro in different color, suspecting that it would make for different quasi-order in each of the regions.  This “regional order” turned out to be less significant than I expected, probably because the stay-stack symmetry (after 4 out-faros) was broken by reversal, and again broken from a partial faro, which extends beyond the range of the reversal.

Mnemonica 00 wheelonly

I then colored the suits differently, with green for clubs (the clover suit), yellow for diamonds (the suit of gold, money and shopping), red for hearts, and blue for spades (black just didn’t look nice with the rest).  Then we fill in the circle, and get a crowded display:

Mnemonica 01 full

Mnemonica in its cyclical order

Splitting them into suits gives much more meaningful representations.  The major pattern is clearly that of “skipping 2” (or modulus-3); this is particularly apparent with the hearts suit, but recur for all other suits as well.  When the “skip 2” pattern holds for multiple suits, for example, between 51-8, the result are 3-block stacks of like colors.

Mnemonica 02 suits

Separation by suits. Note the skip-2 order.

mnemonica-sampleblocks

The minor patterns are two contiguous blocks with mirror symmetries, whose planes of symmetry occurs between 15 (6S) and 16 (5S), and between 25 (5D) and 26 (KD).

mnemonica-mirrorblocks

The two patterns combine to give an interesting quasi-structure to Mnemonica, where beginning from 44 (9C, the point of the partial faro) we get:

Mnemonica 04 blocks

Blocks of like suits.

Or, with values, it becomes:

Mnemonica 05 numberblock

Ordered blocks with values.

Where if we look at the stacks as blocks, they are in turn related by a quasi-mirror symmetry.  Thwarted by the odd-ball 43 (AC)!

Mnemonica 06 overallStructure

Blocking of the blocks

There are no discernable order when the procedure was repeated for values.  The only interesting observation is that every value is circumscribed within a hemi-circle; that is, for every value, there is always a 25 card spacing, in which no cards of that value could be found.  Fantastic, but I don’t think there’s applications to this discovery!

Mnemonica 03 values

Separation by values. The “outside-most bounds” are indicated with a gray line – note that it spans at most 180 degrees.