Jon Awbrey 5:28 pm on January 5, 2026
Tags: Algebra ( 2 ), Arithmetic, Combinatorics ( 2 ), Computation, Graph Theory ( 5 ), Group Theory, logic ( 23 ), Mathematics ( 22 ), Number Theory, Recursion, Representation, Riffs and Rotes, Semiotics ( 19 ), Visualization ( 9 )

Riffs and Rotes • Happy New Year 2026

$\text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}.$

$\begin{array}{llcl} \text{Then} & 2026 & = & 2 \cdot 1013 \\ && = & p_1 p_{170} \\ && = & p_1 p_{2 \cdot 5 \cdot 17} \\ && = & p_1 p_{p_1 p_3 p_7} \\ && = & p_1 p_{p_1 p_{p_2} p_{p_4}} \\ && = & p_1 p_{p_1 p_{p_{p_1}} p_{p_{{p_1}^{p_1}}}} \end{array}$

No information is lost by dropping the terminal 1s. Thus we may write the following form.

$2026 = p p_{p p_{p_p} p_{p_{p^p}}}$

The article linked below tells how forms of that order correspond to a family of digraphs called riffs and a family of graphs called rotes. The riff and rote for $2026$ are shown in the next two Figures.

Riff 2026

Rote 2026

Reference

Riffs and Rotes

cc: Academia.edu • Laws of Form • Ma^thstodon • Research Gate
cc: Cybernetics • Structural Modeling • Systems Science • Syscoi

#algebra, #arithmetic, #combinatorics, #computation, #graph-theory, #group-theory, #logic, #mathematics, #number-theory, #recursion, #representation, #riffs-and-rotes, #semiotics, #visualization

Jon Awbrey 11:30 am on November 15, 2025
Tags: Amphecks ( 2 ), Animata ( 4 ), Boolean Algebra ( 2 ), Boolean Functions ( 3 ), C.S. Peirce ( 21 ), Cactus Graphs ( 3 ), Computational Complexity, Constraint Satisfaction Problems ( 2 ), Differential Logic ( 3 ), Equational Inference ( 3 ), Graph Theory ( 5 ), Group Theory, Laws of Form ( 2 ), logic ( 23 ), Logical Graphs ( 4 ), Mathematics ( 22 ), Minimal Negation Operators ( 3 ), Model Theory ( 2 ), Painted Cacti ( 2 ), Peirce ( 3 ), Proof Theory, Propositional Calculus ( 3 ), Propositional Equation Reasoning Systems ( 2 ), Spencer Brown ( 2 ), Theorem Proving, Visualization ( 9 )

Survey of Animated Logical Graphs • 8

This is a Survey of blog and wiki posts on Logical Graphs, encompassing several families of graph‑theoretic structures originally developed by Charles S. Peirce as graphical formal languages or visual styles of syntax amenable to interpretation for logical applications.

Beginnings

Logical Graphs • First Impressions

Blog Series • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14)

Logical Graphs • Formal Development

Blog Series • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8)

Elements

Examples

Double Negation • (1) • (2) • (3)

Peirce’s Law

This Blog • OEIS Wiki
Blog Series • (1) • (2) • (3) • (4) • (5) • (6) • (7)

Praeclarum Theorema

This Blog • OEIS Wiki
Blog Series • (1) • (2) • (3)

Proof Animations

Blog Series

Alpha Now, Omega Later • (1) • (2) • (3) • (4) • (5) • (6) • (7)

Animated Logical Graphs • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20) • (21) • (22) • (23) • (24) • (25) • (26) • (27) • (28) • (29) • (30) • (31) • (32) • (33) • (34) • (35) • (36) • (37) • (38) • (39) • (40) • (41) • (42) • (43) • (44) • (45) • (46) • (47) • (48) • (49) • (50) • (51) • (52) • (53) • (54) • (55) • (56) • (57) • (58) • (59) • (60) • (61) • (62) • (63) • (64) • (65) • (66) • (67) • (68) • (69) • (70) • (71) • (72) • (73) • (74) • (75) • (76) • (77) • (78) • (79) • (80) • (81)

Cactus Language

Overview • (1) • (2) • (3) • (4)
Preliminaries • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17)
Syntax • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12)
Generalities About Formal Grammars • (1) • (2) • (3)
Stylistics • (1) • (2) • (3) • (4) • (5) • (6)
Pragmatics • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16)
Mechanics • (1) • (2) • (3) • (4) • (5)
Semantics • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8)
Discussions • (1) • (2) • (3)

Logic Syllabus

Discussions • (1) • (2)

Logical Graphs

Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11)

Logical Graphs • Interpretive Duality • (1) • (2) • (3) • (4)

Logical Graphs, Iconicity, Interpretation • (1) • (2)

Logical Graphs, Truth Tables, Venn Diagrams • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9)

Minimal Negation Operators • (1) • (2) • (3) • (4)

Discussions • (1) • (2)

Genus, Species, Pie Charts, Radio Buttons • (1)

Discussions • (1) • (2) • (3) • (4) • (5)

Excursions

Futures Of Logical Graphs
Operator Variables in Logical Graphs • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12)

Discussions • (1) • (2)

Interpretive Duality in Logical Graphs • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8)
Mathematical Duality in Logical Graphs • (1)

Discussions • (1) • (2)

Transformations of Logical Graphs • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14)

Discussions • (1)

Applications

Anamnesis

CSP, GSB, & Me • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20)

cc: FB | Logical Graphs • Laws of Form • Ma^thstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

#amphecks, #animata, #boolean-algebra, #boolean-functions, #c-s-peirce, #cactus-graphs, #computational-complexity, #constraint-satisfaction-problems, #differential-logic, #equational-inference, #graph-theory, #group-theory, #laws-of-form, #logic, #logical-graphs, #mathematics, #minimal-negation-operators, #model-theory, #painted-cacti, #peirce, #proof-theory, #propositional-calculus, #propositional-equation-reasoning-systems, #spencer-brown, #theorem-proving, #visualization

Jon Awbrey 11:48 am on November 9, 2025
Tags: Algebra ( 2 ), Algebra of Logic, C.S. Peirce ( 21 ), Category Theory ( 3 ), Combinatorics ( 2 ), Discrete Mathematics, Duality, Dyadic Relations, Formal Languages ( 2 ), Foundations of Mathematics ( 2 ), Graph Theory ( 5 ), Group Theory, logic ( 23 ), Logic of Relatives ( 14 ), Logical Graphs ( 4 ), Mathematics ( 22 ), Model Theory ( 2 ), Relation Theory ( 15 ), Semiotics ( 19 ), Set Theory, Sign Relational Manifolds ( 3 ), Sign Relations ( 17 ), Triadic Relations ( 16 ), Type Theory ( 2 )

Survey of Relation Theory • 9

In the present Survey of blog and wiki resources for Relation Theory, relations are viewed from the perspective of combinatorics, in other words, as a topic in discrete mathematics, with special attention to finite structures and concrete set‑theoretic constructions, many of which arise quite naturally in applications. This approach to relation theory is distinct from, though closely related to, its study from the perspectives of abstract algebra on the one hand and formal logic on the other.

Elements

Relation Theory

Relational Concepts

Relation Composition	Relation Construction	Relation Reduction
Relative Term	Sign Relation	Triadic Relation
Logic of Relatives	Hypostatic Abstraction	Continuous Predicate

Illustrations

Six Ways of Looking at a Triadic Relation ⌬ 1

Information‑Theoretic Perspective

Mathematical Demonstration and the Doctrine of Individuals • (1) • (2)

Blog Series

Logic of Relatives

Relation Theory • (1) • (2) • (3) • (4) • (5) • (6)

Discussions • (1) • (2) • (3) • (4) • (5)

Relations & Their Relatives • (1) • (2) • (3) • (4)

Comments • (1) • (2) • (3)
Discussions • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12) • (13) • (14) • (15) • (16) • (17) • (18) • (19) • (20) • (21) • (22) • (23) • (24) • (25)