Advanced Mathematical Research Tool
Cayley-Dickson Research Lab
Explore hypercomplex number systems from reals to higher dimensions with interactive visualizations and real-time analytics
Up to 32 Dimensions
From reals to 32-nions
Real-Time Visualization
3D projections & heatmaps
Property Analysis
Automated testing & validation
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About Cayley-Dickson Construction
Each level doubles the dimension by pairing elements: (a, b) with multiplication(a, b)(c, d) = (ac - d̄b, da + bc̄). Properties are progressively lost: commutativity at level 2, associativity at level 3, and alternativity at level 4.
Select Level
Each level doubles the dimension using Cayley-Dickson construction
Current: Level 2 • Quaternions (ℍ) • Dimension: 4
Your Journey
Step 1 of 9
DISCRETE SYSTEMS
Countable, Separated
CONTINUOUS SYSTEMS
Uncountable, Complete
ℕ
Natural Numbers (Set Theory)
The birth of numbers from pure logic! Starting from nothing (∅), we build everything: 0 = {}, 1 = {0} = {{}}, 2 = {0,1}, ...
Discovered by: Giuseppe Peano
Year: 1889
Dimension: 0D
Key Insight
All mathematics builds from nothing! Sets contain sets, and numbers emerge from pure logic.
Number System Properties
Discrete
HOLDS
Numbers are separated, countable, like stepping stones
Well-Ordered
HOLDS
Every non-empty subset has a smallest element
Closed under +×
HOLDS
Adding or multiplying naturals gives naturals
Has Subtraction
FAILS
Cannot compute 3 - 5 in naturals!
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Discrete Number System
Countable, separated values - like beads on a string
Set-Theoretic Construction
0 = {} (empty set)
1 = {{}}} = {0}
2 = {0, 1}
3 = {0, 1, 2}
Each number contains all previous numbers!
Cardinality
All discrete systems share cardinality ℵ₀ (aleph-null) - countably infinite. You can put them in 1-1 correspondence with natural numbers!
💡 Visualization Note: 3D visualizations begin with continuous systems (ℝ and beyond). Discrete systems are best understood through logic and set theory!
Examples
Real-World Applications
Counting
Computer science
Discrete math
Logic
Mathematical Reference
ℝ - Reals
Level 0, dimension 1
A totally ordered field with all operations defined
ℂ - Complex
Level 1, dimension 2
First extension with i² = -1. Commutative, associative field
ℍ - Quaternions
Level 2, dimension 4
Non-commutative but associative division algebra
𝕆 - Octonions
Level 3, dimension 8
Non-associative but alternative division algebra
𝕊 - Sedenions
Level 4, dimension 16
First algebra with zero divisors. Non-alternative
32-nions & Beyond
Level 5+, dimension 32+
Increasingly exotic properties and zero divisors
Each doubling in the Cayley-Dickson construction represents a fundamental trade-off in mathematical structure, illustrating the deep connections between dimension, algebra, and geometry.