Mathematics

_{Focus Areas · Design document §3}

Available from: O05 (Celestial Mathematics) or SO01 (Counting) + SO04 (Writing)

20 breakthroughs — CORE 6 · STANDARD 11 · UNCOMMON 2 · UNIQUE 1 — Ancient – Space

Builds on: Observation (2) · Social Organization (2) · Philosophy and the Arts (3) · Computing (4)

Leads into: Observation (1) · Construction (4) · Warfare (2) · Philosophy and the Arts (3) · Trade (2) · Mechanics (3) · Thermodynamics (2) · Chemistry (1) · Optics (1) · Electricity (2) · Computing (6) · Nuclear Physics (1) · Quintessic Physics (4) · Quantum Physics (6) · Relativity & Cosmology (1) · Rocketry (1) · Aeronautics (2)

_{The numbers count prerequisite links between this focus and the other. How to read the tables.}

Overview

ID	Breakthrough	Tier	Era	Prerequisites
MA01	Geometry	CORE	Ancient	SO01 + O05
MA03	Basic Algebra	CORE	Classical	MA01 (Geometry — place value)
MA05	Calculus	CORE	Early Modern	MA03 + O06 (Natural Philosophy) + PH04 (Scientific Method — the new mechanics it was made for)
MA06	Probability Theory	STANDARD	Early Modern	MA03 + SO11 (Banking — insurance and the merchant's risk)
MA07	Binary Mathematics	CORE	Early Modern	MA01 (Geometry — place value) + PH01 (Formal Logic)
MA08	Non-Euclidean Geometry	CORE	Early Modern	MA01 + MA05
MA09	Statistics	STANDARD	Modern	MA06
MA11	Fractal Mathematics	UNCOMMON	Atomic	MA05 + MA08 + CP01 (Programmable Computer — Mandelbrot's plots)
MA12	Mayan Vigesimal Mathematics	UNIQUE	Ancient	MA01 (Geometry — place value) + Tropical/jungle biome
MA13	Topology	CORE	Modern	MA08 + MA03 (Basic Algebra) + PH01 (Formal Logic)
MA14	Chaos Theory	UNCOMMON	Atomic	MA05 + MA09 + CP01 (Computer)
MA15	Optimisation Theory	STANDARD	Space	MA13 + MA09 + CP01 (Programmable Computer — linear programming ran on computers) — cost ×2.5
MA19	Higher-Dimensional Geometry	STANDARD	Space	MA15 + MA13
MA16	Operations Research	STANDARD	Space	MA15 — cost ×2.5
MA17	Lifecycle Engineering	STANDARD	Space	MA15 — cost ×2.5
MA18	Logistics Optimisation	STANDARD	Space	MA15 — cost ×2.5
MA22	Auction Theory	STANDARD	Space	MA15 — cost ×2.5
MA23	Post-Quantum Cryptography	STANDARD	Space	MA15 + MA07 — cost ×2.5
MA20	Computational Proof	STANDARD	Space	MA15 + CP01 — cost ×2.5
MA21	Automated Mathematics	STANDARD	Space	MA20 + MA19 — cost ×4

Prerequisite map

flowchart LR
  n_MA01["MA01 Geometry"]:::CORE
  n_MA03["MA03 Basic Algebra"]:::CORE
  n_MA05["MA05 Calculus"]:::CORE
  n_MA08["MA08 Non-Euclidean Geometry"]:::CORE
  n_MA11["MA11 Fractal Mathematics"]:::UNCOMMON
  n_MA13["MA13 Topology"]:::CORE
  n_MA15["MA15 Optimisation Theory"]:::STANDARD
  n_MA19["MA19 Higher-Dimensional Geometry"]:::STANDARD
  n_MA21["MA21 Automated Mathematics"]:::STANDARD
  n_MA16["MA16 Operations Research"]:::STANDARD
  n_MA17["MA17 Lifecycle Engineering"]:::STANDARD
  n_MA18["MA18 Logistics Optimisation"]:::STANDARD
  n_MA22["MA22 Auction Theory"]:::STANDARD
  n_MA23["MA23 Post-Quantum Cryptography"]:::STANDARD
  n_MA20["MA20 Computational Proof"]:::STANDARD
  n_MA14["MA14 Chaos Theory"]:::UNCOMMON
  n_MA06["MA06 Probability Theory"]:::STANDARD
  n_MA09["MA09 Statistics"]:::STANDARD
  n_MA07["MA07 Binary Mathematics"]:::CORE
  n_MA12["MA12 Mayan Vigesimal Mathematics"]:::UNIQUE
  n_MA01 --> n_MA03
  n_MA03 --> n_MA05
  n_MA03 --> n_MA06
  n_MA01 --> n_MA07
  n_MA01 --> n_MA08
  n_MA05 --> n_MA08
  n_MA06 --> n_MA09
  n_MA05 --> n_MA11
  n_MA08 --> n_MA11
  n_MA01 --> n_MA12
  n_MA08 --> n_MA13
  n_MA03 --> n_MA13
  n_MA05 --> n_MA14
  n_MA09 --> n_MA14
  n_MA13 --> n_MA15
  n_MA09 --> n_MA15
  n_MA15 --> n_MA19
  n_MA13 --> n_MA19
  n_MA15 --> n_MA16
  n_MA15 --> n_MA17
  n_MA15 --> n_MA18
  n_MA15 --> n_MA22
  n_MA15 --> n_MA23
  n_MA07 --> n_MA23
  n_MA15 --> n_MA20
  n_MA20 --> n_MA21
  n_MA19 --> n_MA21
  classDef CORE fill:#dbeafe,stroke:#1d4ed8,color:#0b1b3b
  classDef STANDARD fill:#f1f5f9,stroke:#64748b,color:#0f172a
  classDef UNCOMMON fill:#dcfce7,stroke:#15803d,color:#052e16
  classDef RARE fill:#fef3c7,stroke:#b45309,color:#3b2005
  classDef UNIQUE fill:#fae8ff,stroke:#a21caf,color:#3b0a3f

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_{Prerequisites within this focus. Box colour is the tier (blue CORE, grey STANDARD, green UNCOMMON, amber RARE, purple UNIQUE); dotted arrows are alternatives (“or”); a line can pass behind another box, so follow it from its start. 9 of these 20 breakthroughs also need breakthroughs from other foci, which the overview above lists.}

Breakthroughs

MA01 · Geometry

• Tier: CORE · Era: Ancient
• Prerequisites: SO01 + O05
• Unlocks: ⟨[-25]% construction time for [All] improvements⟩ — surveyed land is improved faster. Prerequisite for the Arch (CO03) and Mechanics. ⟨[+1 Gold] from all [Market] buildings⟩ — place value keeps the market's accounts: arithmetic and geometry together, as in Babylon and Alexandria (Zero and Place Value, merged here in v3.102).
• Leads to: CO03 Arch Construction, CO20 Earthquake Mastery, WA11 Trebuchet, MA03 Basic Algebra, MA07 Binary Mathematics, MA08 Non-Euclidean Geometry, MA12 Mayan Vigesimal Mathematics, MC03 Gearing, OP02 Lens Grinding, AE07 Stealth Technology

MA03 · Basic Algebra

• Tier: CORE · Era: Classical
• Prerequisites: MA01 (Geometry — place value)
• Unlocks: ⟨[+5]% Production when constructing [All] buildings [in all cities]⟩ — engineering by calculation. ⟨[+10]% Strength <for [Siege] units>⟩ (artillery by angle); ⟨[+1 Culture] from all [World Wonder] buildings⟩ (proportion in stone) — algebra and trigonometry, the mathematics of Alexandria and Baghdad (Trigonometry, merged here in v3.105).
• Leads to: CO06 Flying Buttress, CO13 Geodesic Structures, MA05 Calculus, MA06 Probability Theory, MA13 Topology, TH10 Mechanical Equivalent of Heat, CH06 Periodic Table, AE03 Heavier-Than-Air Flight

MA05 · Calculus

• Tier: CORE · Era: Early Modern
• Prerequisites: MA03 + O06 (Natural Philosophy) + PH04 (Scientific Method — the new mechanics it was made for)
• Unlocks: Unlocks Mechanics focus (advanced); ⟨[+1] Range <for [Siege] units>⟩ (trajectories solved); +15% Insight to the physical foci.
• Leads to: O15 Mechanical Weather Predictor, WA19 Railgun Concept, MA08 Non-Euclidean Geometry, MA11 Fractal Mathematics, MA14 Chaos Theory, TH05 Carnot Efficiency, EL03 Electromagnetism, EL09 Wireless Transmission, CP13 Analog Computing Mastery, NP02 Atomic Model, QM-E01 Fundamental Matter Theory, QP02 Quantum Mechanics Formalism, RK04 Orbital Mechanics

MA06 · Probability Theory

• Tier: STANDARD · Era: Early Modern
• Prerequisites: MA03 + SO11 (Banking — insurance and the merchant's risk)
• Unlocks: ⟨[+1 Gold] from all [Bank] buildings⟩ — insurance priced by probability. ⟨[+10]% [Gold] [in all cities]⟩ — risk, priced and pooled (Insurance, merged here in v3.101).
• Leads to: MA09 Statistics, TR10 Futures Market

MA07 · Binary Mathematics

• Tier: CORE · Era: Early Modern
• Prerequisites: MA01 (Geometry — place value) + PH01 (Formal Logic)
• Unlocks: Cipher Officeⁿ [60/0; ⟨[+25]% spy effectiveness [in this city]⟩ (<when espionage is enabled>)]. Prerequisite for Computing.
• Leads to: MA23 Post-Quantum Cryptography, PH10 Information Theory, MC14 Babbage Mechanical Computer, MC18 Clockwork Analytical Engine, CP01 Programmable Computer

MA08 · Non-Euclidean Geometry

• Tier: CORE · Era: Early Modern
• Prerequisites: MA01 + MA05
• Unlocks: ⟨[+10]% [Gold] from Trade Routes⟩ — accurate maps, shortest paths. Prerequisite for Relativity.
• Leads to: MA11 Fractal Mathematics, MA13 Topology, QM-E04 Unified Field Theory (Elemental), QP10 Loop Quantum Gravity, RC01 Relativity

MA09 · Statistics

• Tier: STANDARD · Era: Modern
• Prerequisites: MA06
• Unlocks: ⟨[+5]% [Production] [in all cities]⟩ — quality control.
• Leads to: MA14 Chaos Theory, MA15 Optimisation Theory, PH11 Systems Thinking, TR10 Futures Market, CP07 Artificial Intelligence (Narrow), CP08 Machine Learning, CP14 Perceptron Machines

MA11 · Fractal Mathematics

• Tier: UNCOMMON · Era: Atomic
• Prerequisites: MA05 + MA08 + CP01 (Programmable Computer — Mandelbrot's plots)
• Unlocks: ⟨[+1 Science] from [Mine] tiles [in all cities]⟩ — terrain read as mathematics.

MA12 · Mayan Vigesimal Mathematics

• Tier: UNIQUE · Era: Ancient
• Prerequisites: MA01 (Geometry — place value) + Tropical/jungle biome
• Unlocks: Long Count Observatoryⁿ [55/1; +2 Science, +1 Faith; +10% Insight to Observation]. Civ unique: ⟨[+1 Science] from [Jungle] tiles [in all cities]⟩ — the calendar is kept in the canopy.
• Notes: Mesoamerican-inspired.

MA13 · Topology

• Tier: CORE · Era: Modern
• Prerequisites: MA08 + MA03 (Basic Algebra) + PH01 (Formal Logic)
• Unlocks: ⟨[+1 Gold, +1 Production] from each Trade Route⟩ — the network optimised. ⟨[-10]% maintenance cost for [All] buildings [in all cities]⟩ — administration by set and rule: Cantor and Poincaré, the new mathematics of the same decades (Set Theory / Formal Logic, merged here in v3.103).
• Leads to: MA15 Optimisation Theory, MA19 Higher-Dimensional Geometry, CP06 Cryptography, QM-E12 Higher Strata Theory, QM-E27 Plenum Gravitation, QP02 Quantum Mechanics Formalism, QP05 Quantum Field Theory, QP09 String Theory Validation, QP13 Hidden Variable Discovery

MA14 · Chaos Theory

• Tier: UNCOMMON · Era: Atomic
• Prerequisites: MA05 + MA09 + CP01 (Computer)
• Unlocks: ⟨[+5]% growth [in all cities]⟩ (long-range forecasting for the fields); ⟨[+10]% to Flank Attack bonuses⟩ (positioning solved).
• Leads to: PH11 Systems Thinking

MA15 · Optimisation Theory

• Tier: STANDARD · Era: Space
• Prerequisites: MA13 + MA09 + CP01 (Programmable Computer — linear programming ran on computers) — cost ×2.5
• Unlocks: ⟨[-15]% maintenance cost for [All] buildings [in all cities]⟩, on top of MA13's −10%.
• Notes: W18 — Entry: opens the pool.
• Leads to: MA19 Higher-Dimensional Geometry, MA16 Operations Research, MA17 Lifecycle Engineering, MA18 Logistics Optimisation, MA22 Auction Theory, MA23 Post-Quantum Cryptography, MA20 Computational Proof

MA19 · Higher-Dimensional Geometry

• Tier: STANDARD · Era: Space
• Prerequisites: MA15 + MA13
• Unlocks: ⟨[+10]% [Science] [in all cities]⟩.
• Notes: W18 — Pool; Hyperspace engineering (both ontologies; the geometry of the strata in elemental worlds).
• Leads to: MA21 Automated Mathematics

MA16 · Operations Research

• Tier: STANDARD · Era: Space
• Prerequisites: MA15 — cost ×2.5
• Unlocks: ⟨[-25]% maintenance costs⟩ (unit upkeep; to confirm against Unciv).
• Notes: W18 — Pool.

MA17 · Lifecycle Engineering

• Tier: STANDARD · Era: Space
• Prerequisites: MA15 — cost ×2.5
• Unlocks: ⟨[-33]% Gold cost of upgrading⟩ — retrofits planned from the start.
• Notes: W18 — Pool.

MA18 · Logistics Optimisation

• Tier: STANDARD · Era: Space
• Prerequisites: MA15 — cost ×2.5
• Unlocks: ⟨[+1] Movement <for [Civilian] units>⟩.
• Notes: W18 — Pool.

MA22 · Auction Theory

• Tier: STANDARD · Era: Space
• Prerequisites: MA15 — cost ×2.5
• Unlocks: ⟨[Gold] cost of purchasing items in cities [-15]%⟩.
• Notes: W18 — Pool.

MA23 · Post-Quantum Cryptography

• Tier: STANDARD · Era: Space
• Prerequisites: MA15 + MA07 — cost ×2.5
• Unlocks: ⟨[+5]% [Gold] [in all cities]⟩ (secure finance); ⟨[-25]% enemy spy effectiveness [in all cities] <when espionage is enabled>⟩.
• Notes: W18 — Pool.

MA20 · Computational Proof

• Tier: STANDARD · Era: Space
• Prerequisites: MA15 + CP01 — cost ×2.5
• Unlocks: ⟨[-10]% Science cost of researching new Technologies⟩.
• Notes: W18 — First step of capstone A.
• Leads to: MA21 Automated Mathematics

MA21 · Automated Mathematics

• Tier: STANDARD · Era: Space
• Prerequisites: MA20 + MA19 — cost ×4
• Unlocks: ⟨[-15]% Science cost of researching new Technologies⟩, ⟨[+10]% [Science] [in all cities]⟩.
• Notes: W18 — Capstone A.