Mathematical Background¶

Vigesimal Number System¶

Base-20 Decomposition¶

Any non-negative integer \(n\) can be represented in base 20 as:

\[n = \sum_{i=0}^{L-1} d_i \cdot 20^i\]

where \(d_i \in \{0, 1, \ldots, 19\}\) are the vigesimal digits and \(L\) is the number of levels.

Bars and Dots¶

Each vigesimal digit \(d\) (0-19) is further decomposed:

\[\text{bars}(d) = \lfloor d / 5 \rfloor \in \{0, 1, 2, 3\}\]

\[\text{dots}(d) = d \mod 5 \in \{0, 1, 2, 3, 4\}\]

This creates a two-level hierarchy within each digit position.

Feature Vector¶

For components="full" with \(L\) levels, each scalar produces a \(3L\)-dimensional feature vector:

\[\mathbf{f}(n) = [d_0, b_0, \dot{d}_0, d_1, b_1, \dot{d}_1, \ldots, d_{L-1}, b_{L-1}, \dot{d}_{L-1}]\]

Maya Calendar Mathematics¶

Tzolk'in Cycle¶

The Tzolk'in position is determined by two independent cycles:

\[\text{number} = ((J - J_0) \mod 13) + 1\]

\[\text{name\_index} = (J - J_0) \mod 20\]

where \(J\) is the Julian Day Number and \(J_0 = 584283\) is the GMT correlation epoch.

Since \(\gcd(13, 20) = 1\), the combined cycle length is \(13 \times 20 = 260\) days (by the Chinese Remainder Theorem).

Haab' Calendar¶

The Haab' position within its 365-day cycle:

\[\text{day\_of\_year} = (J - J_0) \mod 365\]

\[\text{month} = \lfloor \text{day\_of\_year} / 20 \rfloor\]

\[\text{day\_of\_month} = \text{day\_of\_year} \mod 20\]

The 5 Wayeb' days occur when month = 18 (the 19th "month").

Long Count (Mixed Radix)¶

The Long Count uses a modified vigesimal system:

\[J - J_0 = k_0 + 20 \cdot k_1 + 360 \cdot k_2 + 7200 \cdot k_3 + 144000 \cdot k_4\]

The multipliers are:

Level	Name	Radix	Cumulative
0	K'in	20	1
1	Uinal	18	20
2	Tun	20	360
3	K'atun	20	7,200
4	B'ak'tun	20	144,000

Note the exception at level 1 → 2: the multiplier is 18 (not 20) so that a Tun ≈ 360 days, approximating the solar year.

Cyclical Encoding¶

For a component with period \(P\) and value \(v\):

\[\text{sin} = \sin\left(\frac{2\pi v}{P}\right), \quad \text{cos} = \cos\left(\frac{2\pi v}{P}\right)\]

This maps the cyclic variable to a unit circle, ensuring \(v = 0\) and \(v = P\) map to the same point, which is important for smooth feature transitions at cycle boundaries.

Calendar Round¶

The Calendar Round combines Tzolk'in (260 days) and Haab' (365 days):

\[\text{CR period} = \text{lcm}(260, 365) = 18{,}980 \text{ days} \approx 52 \text{ years}\]

This 52-year cycle was the longest period the Maya could reference without the Long Count.