Marmeid diagram

.pre-commit-config.yaml

@@ -11,6 +11,7 @@ repos:
   - id: check-merge-conflict
   - id: check-xml
   - id: check-yaml
+    args: ['--unsafe']
   - id: debug-statements
   - id: end-of-file-fixer
   - id: requirements-txt-fixer

docs/assets/js/mathjax.js

@@ -1,9 +1,11 @@
 window.MathJax = {
+    loader: {load: ['[tex]/colortbl']},
     tex: {
       inlineMath: [["\\(", "\\)"]],
       displayMath: [["\\[", "\\]"]],
       processEscapes: true,
-      processEnvironments: true
+      processEnvironments: true,
+      packages: {'[+]': ['colortbl']},
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docs/fundamentals/ideas/congeneric_decomposition.md

@@ -1,3 +1,220 @@
+---
+hide:
+  - toc
+---
 # Congeneric Decomposition
 
-Coming soon
+Cogeneric decomposition is a method for decomposing symbolic sequences from a systems thinking perspective,
+emphasizing the importance of order. It decomposes a sequence into a tuple of cogeneric sequences,
+each of which consists of equivalent elements at certain positions, while all other positions are empty.
+This reversible process preserves the order of the sequence and allows the original sequence to be fully reconstructed.
+
+The concept of Cogeneric decomposition can be demonstrated using an example:
+
+Let's assume there is a symbolic sequence `INTELLIGENCE IS THE ABILITY TO ADAPT` congeneric decomposition
+could be presented by the following table, where each row is a congeneric sequence and  `-` is an empty position in a congeneric sequence.
+
+
+<!-- \begin{equation}
+\scriptsize
+\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
+\hline
+\cellcolor{#ff7f0e}I & \cellcolor{#ffbb78}N & \cellcolor{#2ca02c}T & \cellcolor{#98df8a}E & \cellcolor{#d62728}L & \cellcolor{#d62728}L & \cellcolor{#ff7f0e}I & \cellcolor{#ff9896}G & \cellcolor{#98df8a}E & \cellcolor{#ffbb78}N & \cellcolor{#9467bd}C & \cellcolor{#98df8a}E & \cellcolor{#c5b0d5}\text{ } & \cellcolor{#ff7f0e}I & \cellcolor{#8c564b}S & \cellcolor{#c5b0d5}\text{ } & \cellcolor{#2ca02c}T & \cellcolor{#c49c94}H & \cellcolor{#98df8a}E & \cellcolor{#c5b0d5}\text{ } & \cellcolor{#e377c2}A & \cellcolor{#f7b6d2}B & \cellcolor{#ff7f0e}I & \cellcolor{#d62728}L & \cellcolor{#ff7f0e}I & \cellcolor{#2ca02c}T & \cellcolor{#bcbd22}Y & \cellcolor{#c5b0d5}\text{ } & \cellcolor{#2ca02c}T & \cellcolor{#dbdb8d}O & \cellcolor{#c5b0d5}\text{ } & \cellcolor{#e377c2}A & \cellcolor{#17becf}D & \cellcolor{#e377c2}A & \cellcolor{#9edae5}P & \cellcolor{#2ca02c}T \\
+\hline
+\cellcolor{#ff7f0e}I & - & - & - & - & - & \cellcolor{#ff7f0e}I & - & - & - & - & - & - & \cellcolor{#ff7f0e}I & - & - & - & - & - & - & - & - & \cellcolor{#ff7f0e}I & - & \cellcolor{#ff7f0e}I & - & - & - & - & - & - & - & - & - & - & - \\
+\hline
+- & \cellcolor{#ffbb78}N & - & - & - & - & - & - & - & \cellcolor{#ffbb78}N & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\
+\hline
+- & - & \cellcolor{#2ca02c}T & - & - & - & - & - & - & - & - & - & - & - & - & - & \cellcolor{#2ca02c}T & - & - & - & - & - & - & - & - & \cellcolor{#2ca02c}T & - & - & \cellcolor{#2ca02c}T & - & - & - & - & - &- & \cellcolor{#2ca02c}T \\
+\hline
+- & - & - & \cellcolor{#98df8a}E & - & - & - & - & \cellcolor{#98df8a}E & - & - & \cellcolor{#98df8a}E & - & - & - & - & - & - & \cellcolor{#98df8a}E & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\
+\hline
+- & - & - & - & \cellcolor{#d62728}L & \cellcolor{#d62728}L & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & \cellcolor{#d62728}L & - & - & - & - & - & - & - & - & - & - & - & - \\
+\hline
+- & - & - & - & - & - & - & \cellcolor{#ff9896}G & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\
+\hline
+- & - & - & - & - & - & - & - & - & - & \cellcolor{#9467bd}C & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\
+\hline
+- & - & - & - & - & - & - & - & - & - & - & - & \cellcolor{#c5b0d5}\text{ } & - & - & \cellcolor{#c5b0d5}\text{ } & - & - & - & \cellcolor{#c5b0d5}\text{ } & - & - & - & - & - & - & - & \cellcolor{#c5b0d5}\text{ } & - & - & \cellcolor{#c5b0d5}\text{ } & - & - & - & - & - \\
+\hline
+- & - & - & - & - & - & - & - & - & - & - & - & - & - & \cellcolor{#8c564b}S & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\
+\hline
+- & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & \cellcolor{#c49c94}H & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\
+\hline
+- & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & \cellcolor{#e377c2}A & - & - & - & - & - & - & - & - & - & - & \cellcolor{#e377c2}A & - & \cellcolor{#e377c2}A & - & - \\
+\hline
+- & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & \cellcolor{#f7b6d2}B & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\
+\hline
+- & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & \cellcolor{#bcbd22}Y & - & - & - & - & - & - & - & - & - \\
+\hline
+- & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & \cellcolor{#dbdb8d}O & - & - & - & - & - & - \\
+\hline
+- & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & \cellcolor{#17becf}D & - & - & - \\
+\hline
+- & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & \cellcolor{#9edae5}P & - \\
+\hline
+\end{array}
+\end{equation}
+ -->
+
+
+
+``` mermaid
+block-beta
+  columns 36
+  seq1["I"] seq2["N"] seq3["T"] seq4["E"] seq5["L"] seq6["L"] seq7["I"] seq8["G"] seq9["E"] seq10["N"]
+  seq11["C"] seq12["E"] seq13[" "] seq14["I"] seq15["S"] seq16[" "] seq17["T"] seq18["H"] seq19["E"] seq20[" "]
+  seq21["A"] seq22["B"] seq23["I"] seq24["L"] seq25["I"] seq26["T"] seq27["Y"] seq28[" "] seq29["T"] seq30["O"]
+  seq31[" "] seq32["A"] seq33["D"] seq34["A"] seq35["P"] seq36["T"]
+
+  space:36
+
+  i1["I"] i2["-"] i3["-"] i4["-"] i5["-"] i6["-"] i7["I"] i8["-"] i9["-"] i10["-"]
+  i11["-"] i12["-"] i13["-"] i14["I"] i15["-"] i16["-"] i17["-"] i18["-"] i19["-"] i20["-"]
+  i21["-"] i22["-"] i23["I"] i24["-"] i25["I"] i26["-"] i27["-"] i28["-"] i29["-"] i30["-"]
+  i31["-"] i32["-"] i33["-"] i34["-"] i35["-"] i36["-"]
+
+  n1["-"] n2["N"] n3["-"] n4["-"] n5["-"] n6["-"] n7["-"] n8["-"] n9["-"] n10["N"]
+  n11["-"] n12["-"] n13["-"] n14["-"] n15["-"] n16["-"] n17["-"] n18["-"] n19["-"] n20["-"]
+  n21["-"] n22["-"] n23["-"] n24["-"] n25["-"] n26["-"] n27["-"] n28["-"] n29["-"] n30["-"]
+  n31["-"] n32["-"] n33["-"] n34["-"] n35["-"] n36["-"]
+
+  t1["-"] t2["-"] t3["T"] t4["-"] t5["-"] t6["-"] t7["-"] t8["-"] t9["-"] t10["-"]
+  t11["-"] t12["-"] t13["-"] t14["-"] t15["-"] t16["-"] t17["T"] t18["-"] t19["-"] t20["-"]
+  t21["-"] t22["-"] t23["-"] t24["-"] t25["-"] t26["T"] t27["-"] t28["-"] t29["T"] t30["-"]
+  t31["-"] t32["-"] t33["-"] t34["-"] t35["-"] t36["T"]
+
+  e1["-"] e2["-"] e3["-"] e4["E"] e5["-"] e6["-"] e7["-"] e8["-"] e9["E"] e10["-"]
+  e11["-"] e12["E"] e13["-"] e14["-"] e15["-"] e16["-"] e17["-"] e18["-"] e19["E"] e20["-"]
+  e21["-"] e22["-"] e23["-"] e24["-"] e25["-"] e26["-"] e27["-"] e28["-"] e29["-"] e30["-"]
+  e31["-"] e32["-"] e33["-"] e34["-"] e35["-"] e36["-"]
+
+  l1["-"] l2["-"] l3["-"] l4["-"] l5["L"] l6["L"] l7["-"] l8["-"] l9["-"] l10["-"]
+  l11["-"] l12["-"] l13["-"] l14["-"] l15["-"] l16["-"] l17["-"] l18["-"] l19["-"] l20["-"]
+  l21["-"] l22["-"] l23["-"] l24["L"] l25["-"] l26["-"] l27["-"] l28["-"] l29["-"] l30["-"]
+  l31["-"] l32["-"] l33["-"] l34["-"] l35["-"] l36["-"]
+
+  g1["-"] g2["-"] g3["-"] g4["-"] g5["-"] g6["-"] g7["-"] g8["G"] g9["-"] g10["-"]
+  g11["-"] g12["-"] g13["-"] g14["-"] g15["-"] g16["-"] g17["-"] g18["-"] g19["-"] g20["-"]
+  g21["-"] g22["-"] g23["-"] g24["-"] g25["-"] g26["-"] g27["-"] g28["-"] g29["-"] g30["-"]
+  g31["-"] g32["-"] g33["-"] g34["-"] g35["-"] g36["-"]
+
+  c1["-"] c2["-"] c3["-"] c4["-"] c5["-"] c6["-"] c7["-"] c8["-"] c9["-"] c10["-"]
+  c11["C"] c12["-"] c13["-"] c14["-"] c15["-"] c16["-"] c17["-"] c18["-"] c19["-"] c20["-"]
+  c21["-"] c22["-"] c23["-"] c24["-"] c25["-"] c26["-"] c27["-"] c28["-"] c29["-"] c30["-"]
+  c31["-"] c32["-"] c33["-"] c34["-"] c35["-"] c36["-"]
+
+  sp1["-"] sp2["-"] sp3["-"] sp4["-"] sp5["-"] sp6["-"] sp7["-"] sp8["-"] sp9["-"] sp10["-"]
+  sp11["-"] sp12["-"] sp13[" "] sp14["-"] sp15["-"] sp16[" "] sp17["-"] sp18["-"] sp19["-"] sp20[" "]
+  sp21["-"] sp22["-"] sp23["-"] sp24["-"] sp25["-"] sp26["-"] sp27["-"] sp28[" "] sp29["-"] sp30["-"]
+  sp31[" "] sp32["-"] sp33["-"] sp34["-"] sp35["-"] sp36["-"]
+
+  s1["-"] s2["-"] s3["-"] s4["-"] s5["-"] s6["-"] s7["-"] s8["-"] s9["-"] s10["-"]
+  s11["-"] s12["-"] s13["-"] s14["-"] s15["S"] s16["-"] s17["-"] s18["-"] s19["-"] s20["-"]
+  s21["-"] s22["-"] s23["-"] s24["-"] s25["-"] s26["-"] s27["-"] s28["-"] s29["-"] s30["-"]
+  s31["-"] s32["-"] s33["-"] s34["-"] s35["-"] s36["-"]
+
+  h1["-"] h2["-"] h3["-"] h4["-"] h5["-"] h6["-"] h7["-"] h8["-"] h9["-"] h10["-"]
+  h11["-"] h12["-"] h13["-"] h14["-"] h15["-"] h16["-"] h17["-"] h18["H"] h19["-"] h20["-"]
+  h21["-"] h22["-"] h23["-"] h24["-"] h25["-"] h26["-"] h27["-"] h28["-"] h29["-"] h30["-"]
+  h31["-"] h32["-"] h33["-"] h34["-"] h35["-"] h36["-"]
+
+  a1["-"] a2["-"] a3["-"] a4["-"] a5["-"] a6["-"] a7["-"] a8["-"] a9["-"] a10["-"]
+  a11["-"] a12["-"] a13["-"] a14["-"] a15["-"] a16["-"] a17["-"] a18["-"] a19["-"] a20["-"]
+  a21["A"] a22["-"] a23["-"] a24["-"] a25["-"] a26["-"] a27["-"] a28["-"] a29["-"] a30["-"]
+  a31["-"] a32["A"] a33["-"] a34["A"] a35["-"] a36["-"]
+
+  b1["-"] b2["-"] b3["-"] b4["-"] b5["-"] b6["-"] b7["-"] b8["-"] b9["-"] b10["-"]
+  b11["-"] b12["-"] b13["-"] b14["-"] b15["-"] b16["-"] b17["-"] b18["-"] b19["-"] b20["-"]
+  b21["-"] b22["B"] b23["-"] b24["-"] b25["-"] b26["-"] b27["-"] b28["-"] b29["-"] b30["-"]
+  b31["-"] b32["-"] b33["-"] b34["-"] b35["-"] b36["-"]
+
+  y1["-"] y2["-"] y3["-"] y4["-"] y5["-"] y6["-"] y7["-"] y8["-"] y9["-"] y10["-"]
+  y11["-"] y12["-"] y13["-"] y14["-"] y15["-"] y16["-"] y17["-"] y18["-"] y19["-"] y20["-"]
+  y21["-"] y22["-"] y23["-"] y24["-"] y25["-"] y26["-"] y27["Y"] y28["-"] y29["-"] y30["-"]
+  y31["-"] y32["-"] y33["-"] y34["-"] y35["-"] y36["-"]
+
+  o1["-"] o2["-"] o3["-"] o4["-"] o5["-"] o6["-"] o7["-"] o8["-"] o9["-"] o10["-"]
+  o11["-"] o12["-"] o13["-"] o14["-"] o15["-"] o16["-"] o17["-"] o18["-"] o19["-"] o20["-"]
+  o21["-"] o22["-"] o23["-"] o24["-"] o25["-"] o26["-"] o27["-"] o28["-"] o29["-"] o30["O"]
+  o31["-"] o32["-"] o33["-"] o34["-"] o35["-"] o36["-"]
+
+  d1["-"] d2["-"] d3["-"] d4["-"] d5["-"] d6["-"] d7["-"] d8["-"] d9["-"] d10["-"]
+  d11["-"] d12["-"] d13["-"] d14["-"] d15["-"] d16["-"] d17["-"] d18["-"] d19["-"] d20["-"]
+  d21["-"] d22["-"] d23["-"] d24["-"] d25["-"] d26["-"] d27["-"] d28["-"] d29["-"] d30["-"]
+  d31["-"] d32["-"] d33["D"] d34["-"] d35["-"] d36["-"]
+
+  p1["-"] p2["-"] p3["-"] p4["-"] p5["-"] p6["-"] p7["-"] p8["-"] p9["-"] p10["-"]
+  p11["-"] p12["-"] p13["-"] p14["-"] p15["-"] p16["-"] p17["-"] p18["-"] p19["-"] p20["-"]
+  p21["-"] p22["-"] p23["-"] p24["-"] p25["-"] p26["-"] p27["-"] p28["-"] p29["-"] p30["-"]
+  p31["-"] p32["-"] p33["-"] p34["-"] p35["P"] p36["-"]
+
+  classDef c1 fill:#ff7f0e,color:#fff;
+  classDef c2 fill:#ffbb78,color:#000;
+  classDef c3 fill:#2ca02c,color:#fff;
+  classDef c4 fill:#98df8a,color:#000;
+  classDef c5 fill:#d62728,color:#fff;
+  classDef c6 fill:#ff9896,color:#000;
+  classDef c7 fill:#9467bd,color:#fff;
+  classDef c8 fill:#c5b0d5,color:#000;
+  classDef c9 fill:#8c564b,color:#fff;
+  classDef c10 fill:#c49c94,color:#000;
+  classDef c11 fill:#e377c2,color:#fff;
+  classDef c12 fill:#f7b6d2,color:#000;
+  classDef c13 fill:#bcbd22,color:#fff;
+  classDef c14 fill:#dbdb8d,color:#000;
+  classDef c15 fill:#17becf,color:#fff;
+  classDef c16 fill:#9edae5,color:#000;
+
+  class seq1,seq7,seq14,seq23,seq25,i1,i7,i14,i23,i25 c1
+  class seq2,seq10,n2,n10 c2
+  class seq3,seq17,seq26,seq29,seq36,t3,t17,t26,t29,t36 c3
+  class seq4,seq9,seq12,seq19,e4,e9,e12,e19  c4
+  class seq5,seq6,seq24,l5,l6,l24 c5
+  class seq8,g8 c6
+  class seq11,c11 c7
+  class seq13,seq16,seq20,seq28,seq31,sp13,sp16,sp20,sp28,sp31 c8
+  class seq15,s15 c9
+  class seq18,h18 c10
+  class seq21,seq32,seq34,a21,a32,a34 c11
+  class seq22,b22 c12
+  class seq27,y27 c13
+  class seq30,o30 c14
+  class seq33,d33 c15
+  class seq35,p35 c16
+```
+
+Congeneric sequence for `E`
+``` mermaid
+block-beta
+  columns 36
+  e1["-"] e2["-"] e3["-"] e4["E"] e5["-"] e6["-"] e7["-"] e8["-"] e9["E"] e10["-"]
+  e11["-"] e12["E"] e13["-"] e14["-"] e15["-"] e16["-"] e17["-"] e18["-"] e19["E"] e20["-"]
+  e21["-"] e22["-"] e23["-"] e24["-"] e25["-"] e26["-"] e27["-"] e28["-"] e29["-"] e30["-"]
+  e31["-"] e32["-"] e33["-"] e34["-"] e35["-"] e36["-"]
+```
+
+
+could be a part of multiple symbol sequences that have the same order of `E` element.
+
+While keeping the main idea, the congeneric decomposition could be applied, with a flavor, to any type of special case symbolic sequences, such as Order.
+
+<!-- TODO: Add example of congeneric decomposition code -->
+
+<style>
+.md-typeset table:not([class]) th {
+    min-width: 0 !important;
+}
+
+.md-typeset td:not([class]):not(:last-child), .md-typeset th:not([class]):not(:last-child) {
+    border-right: .05rem solid var(--md-typeset-table-color);
+}
+
+.md-typeset td, .md-typeset th {
+    padding-left: 0.4em !important;
+    padding-right: 0.4em !important;
+    padding-top: 0.1em !important;
+    padding-bottom: 0.1em !important;
+    text-align: center !important;
+}
+</style>

docs/fundamentals/ideas/geometric_mean_based_characteristics.md

@@ -1,3 +1,40 @@
+---
+hide:
+  - toc
+---
 # Geometric Mean as Alternative to Probability
 
-Coming soon
+At first glance, introducing the concept of [intervals](interval_as_a_basic_information_unit.md) may seem like an unnecessary complication.
+After all, if the ultimate goal is to estimate the probability of a symbol, there are much simpler methods - counting the frequency of occurrence of a symbol relative to the total.
+
+However, this perspective begins to shift when we consider other types of aggregate functions beyond the arithmetic mean.
+One particularly insightful example is the geometric mean of intervals.
+While the arithmetic mean smooths out the "structure" of the data and bring us back to probability
+(since the average interval between identical symbols is simply the inverse of their probability),
+the geometric mean responds to the diversity of intervals in a fundamentally different way.
+
+If the intervals between repeated elements are uniform, the geometric mean and the arithmetic mean will be the same.
+But as the intervals become more irregular — because symbols appear in bursts or clusters — [the geometric mean begins
+to diverge from the arithmetic mean](https://en.wikipedia.org/wiki/AM%E2%80%93GM_inequality).
+This makes it a sensitive indicator of the order within the sequence, not just the frequency.
+
+![AM-GM inequality visual proof](https://upload.wikimedia.org/wikipedia/commons/d/d9/AM_GM_inequality_visual_proof.svg)
+
+*Visual proof of the arithmetic mean - geometric mean inequality. Source: [wikipedia.org](https://en.wikipedia.org/wiki/File:AM_GM_inequality_visual_proof.svg)*
+
+Building on this idea, Former Order Analysis explored the potential of reinterpreting classical probabilistic and information-theoretic measures in terms of these intervals.
+Instead of relying solely on symbol frequencies, it reformulated the measures using the geometric mean instead of probability (arithmetic mean).
+
+\begin{array}{|c|c|}
+\hline
+Entropy & Average \ remoteness \\
+\hline
+H= - \sum_{j=1}^{m}{p_j \log_2{p_j}} = \frac {1} {n} * \sum_{j=1}^{m}{n_j \log_2 \Delta_{a_j}} & g = \frac{1}{n} * \sum_{j=1}^{m}{n_j \log_2{\Delta_{g_j}}} = \frac{1}{n} * \sum_{j=1}^{m}{\sum_{i=1}^{n_j} \log_2 \Delta_{ij}} \\
+\hline
+\end{array}
+
+*Example of Shennon's Entropy analog - Average remoteness. Where $n$ - seqeunce length, $m$ - alphabet power, $n_j$ - count of element $j$-th, $\Delta_{a_j}$ - average mean of intervals for element $j$-th, $\Delta_{g_j}$ - geometric mean of intervals for element $j$-th, $\Delta_{ij}$ - $i$-th interval for element $j$-thg*
+
+These measures are fine-grained and sensitive to the temporal or spatial order of elements in a sequence.
+Allows us to distinguish between sequences of symbols that may have identical probability distributions
+but differ in the way those symbols are arranged - insight that traditional measures completely miss.

docs/fundamentals/ideas/index.md

@@ -1,3 +1,20 @@
 # Ideas
 
-Coming soon
+Formal Order Analsis is based on five main ideas:
+
+* [Sequence as a Whole Object](sequence_as_a_whole_object.md) -
+Treat a symbolic sequence as a unique whole, focusing on its internal structure and emergent properties.
+
+* [Order as a Property](order_as_a_property.md) -
+By replacing each symbol in a sequence with its index in a dynamic alphabet, we define an "Order" — a new property that separates the content of the sequence from its composition.
+
+* [Congeneric Decomposition](congeneric_decomposition.md) -
+This method breaks a symbolic sequence into layered sub-sequences where each layer isolates a single symbol’s positions,
+preserving the sequence’s internal order and making its structure more analyzable.
+
+* [Interval as a Basic Information Unit](interval_as_a_basic_information_unit.md) -
+Intervals measure the distance between repeated elements, revealing hidden patterns in repetition and spacing.
+
+* [Geometric Mean as Alternative to Probability](geometric_mean_based_characteristics.md) -
+Using measurements based on the geometric mean of intervals instead of the probability of symbols (the arithmetic mean of occurrence)
+allows us to move from recording simple frequency to recording the underlying structure and regularity in the arrangement of symbols.

docs/fundamentals/ideas/interval_as_a_basic_information_unit.md

@@ -1,3 +1,54 @@
+---
+hide:
+  - toc
+---
 # Interval as a Basic Information Unit
 
-Coming soon
+Intervals serve as a fundamental unit of information by measuring the number of different
+items, events, or symbols that occur between reseated in a sequence.
+
+The intervals for symbol `A` in the following sequence would be `[3, 3, 1, 1, 2, 1, 1]`
+``` mermaid
+block-beta
+  columns 12
+  s1["A"] s2["C"] s3["T"] s4["A"] s5["C"] s6["G"] s7["A"] s8["A"] s9["A"] s10["T"] s11["A"] s12["A"]
+  i1["3"]:3 i2["3"]:3 i3["1"]:1 i4["1"]:1 i5["2"]:2 i6["1"]:1 i7["1"]:1
+
+  classDef c3 fill:#2ca02c,color:#fff;
+  classDef c4 fill:#98df8a,color:#000;
+  class s1,s4,s7,s8,s9,s11,s12 c3
+  class i1,i2,i3,i4,i5,i6,i7 c4
+```
+
+In general, a sequence does not necessarily end with the same symbol it begins with.
+To cover all cases, we consider the sequence as a looped sequence representing an infinite pattern with the same characteristics as the original data
+This cyclic approach corresponds to the idea of ​​representativeness heuristic.
+
+The intervals for symbol `C` in the following cycled sequence would be `[3, 9]`
+``` mermaid
+block-beta
+  columns 15
+  s1["A"] s2["C"] s3["T"] s4["A"] s5["C"] s6["G"] s7["A"] s8["A"] s9["A"] s10["T"] s11["A"] s12["A"] space s13["T"] s14["C"]
+  space i1["3"]:3 i2["9"]:10
+  s12 --> s13
+
+  classDef c3 fill:#2ca02c,color:#fff;
+  classDef c4 fill:#98df8a,color:#000;
+  class s2,s5,s14 c3
+  class i1,i2 c4
+```
+
+The circular pattern preserves both the statistical properties and the order of elements.
+Moreover, the average interval length is the inverse of the probability of an event, which directly relates intervals to probability.
+
+\begin{array}{|c|c|c|}
+\hline
+ & \Delta_a  & P \\
+\hline
+A & \frac{3 + 3 + 1 + 1 + 2 + 1 + 1}{7} = \frac{12}{7} \approx 1.7142; & \frac{7}{12} = (\frac{12}{7})^{-1} = \Delta_a^{-1} \\
+\hline
+C & \frac{3 + 9}{2} = \frac{12}{2} = 6 & \frac{2}{12} = \frac{1}{6} = 6^{-1} = \Delta_a^{-1} \\
+\hline
+\end{array}
+
+This makes intervals a crucial informational unit that offers deeper insights into the sequence than individual occurrences alone.