"Unlock the branches of mathematics and learn how numbers offer a new perspective on the world!" ― Information Tab
Beautiful Complexity is a limited time exploration which focuses on Mathematics. It features 8 generators and 61 upgrades that produce either Real Numbers 
or Imaginary Numbers 
.
Story[]
Opening[]
"A world without numbers is a world of confusion. Nothing can be measured. Nothing can be built. What is this framework through which the world is understood?"
Ending[]
"Math creates order and understanding. It enables the creation of technologies and cultures. Math lets us experience the beauty of complexity."
Objectives & Rewards[]
The requirements that have to be completed in order to get all rewards.
Explore Mathematics (12 Requirements)
- Collect One → 1
- Collect Fibonacci Sequence → 2
- Collect Integers → 3
- Collect Algebra → Algebra Badge
- Collect Quadratic Formula, Proofs → 4
- Collect Pythagorean Theorem → 5
- Collect Fractals, Voronoi Pattern → 7
- Collect Pi → Pi Badge
- Collect Statistics, Fermat's Last Theorem → 8
- Collect Proof by Contradiction, Linear Algebra → 9
- Collect Proof by Induction, Arrow's Impossibility → 11
- Collect The Most Beautiful Equation → Euler's Identity Badge
Badges[]
This exploration holds some rewards already mentioned above. The main ones being these three badges: Bronze: Algebra, Silver: Pi and Gold: Euler's Identity which have an effect on all other evolutionary branches, speeding up every simulation by 1%, and also speeding up production in future Beautiful Complexity simulations by 5, 10 and 15% respectively.
| Image | Name | Description |
|---|---|---|
 |
Algebra |
|
 |
Pi |
|
 |
Euler's Identity |
|
Generators[]
| Icon | Name | Description | Base Cost | Base Production | Requires |
|---|---|---|---|---|---|
|
Building Blocks | Exploring the relationships between numbers is the foundation of mathematics. As a discipline, math is as old as civilization. Mathematical calculations can be found in Babylonian tablets and Egyptian hieroglyphs. |
200
|
1/sec
|
Numbers |
|
Arithmetic | Numbers interact with each other through operations. These operations are studied through arithmetic. Expanding on prehistoric developments, early civilizations build numerical systems to quantify a wide range of information. |
1,500
|
5/sec
|
Building Blocks |
|
Algebra | Building on arithmetic, algebra introduces the idea of unknown values along with methods for determining these values. In Medieval Persia, Al-Khwarizmi introduces a method to solve quadratics, marking the start of modern algebra. |
1e8
|
20,000/sec
|
Multiplication |
|
Applied Math | Numbers and mathematics are a foundational technology for human culture. They facilitate the development of cities, allow Romans to build aqueducts that stand for centuries, and make it possible for humans to travel into space. |
5e9
|
2e6/sec
|
Exponentiation |
|
Geometry | The study of shapes and measurements, geometry can visually express distances, spaces, and figures. One of the first fields of mathematics, geometry allows for the construction of obelisks and pyramids without modern notation. |
5e12
|
2e9/sec
|
Pascal's Triangle |
|
Marvels and Mysteries | Math can prove supposed impossibilities, while seemingly simple problems are unsolved to this day. There are hundreds of problems that remain unanswered, with mathematicians actively working to solve them. |
1e16
|
1/sec
|
Complex Numbers |
|
Calculus | Whether it is applied to the infinitely small or the infinitely large, calculus studies rates of continuous change. Co-credited with its invention, Newton uses calculus to measure planetary orbits and develop the laws of motion. |
1e7
|
2,000/sec
|
i Irrationals |
|
Discrete Math | Where calculus focuses on continuous solutions, discrete math studies problems with individual elements like sets and combinations. Discrete math is the foundation of computer science and software development. |
2e11
|
1e17/sec
|
Variables Statistics |
Upgrades[]
Miscellaneous Effects[]
| Icon | Name | Description | Cost | Effect | Requires |
|---|---|---|---|---|---|
|
One | Math is built on the concept of counting, starting with 1, a single object. With multiple objects, there are multiple 'ones.' Objects can be tracked with tally marks, which Paleolithic humans likely used to count. |
1
|
+0.99 per Real Numbers tap | - |
|
Numbers | Tallying might work for small sums, but not for larger groups of objects. In 6th century India, mathematicians build a system of numbers based on ten symbols. The Hindu-Arabic numerals are 1, 2, 3, 4, 5, 6, 7, 8, 9...and 0. |
100
|
+3 per Real Numbers tap | One |
Building Blocks Efficiency[]
Building Blocks has 10 upgrades, increasing the generator efficiency with a total x1.09601e21 multiplier.
| Icon | Name | Description | Cost | Efficiency | Requires |
|---|---|---|---|---|---|
|
Zero | The absence of a quantity of items can be represented with the numeral 0. Mesopotamians depict zero as early as 300 BCE. Centuries later, Mayan civilizations develop it independently as a placeholder within their numeral system. |
750
|
100% | Building Blocks |
|
Negatives | Numbers that have a value of less than 0 are called negative. 4 - 9 = -5. In 200 BCE, Chinese officials calculate taxes with a system using red rods for positive numbers and black rods for negatives. |
500,000
|
8,000% | Addition |
|
Integers | All of the counting numbers, including their negative counterparts and zero, are called integers. The integers are the set {..., -3, -2, -1, 0, 1, 2, 3, ...}. |
5e6
|
200% | Numbers Negatives |
|
Rationals | Rational numbers are those that can be written as a "fraction", meaning an integer divided by another integer. When turned into decimals, rational numbers either end or repeat. 4.5 is 9/2, and 1/3 is 0.333... |
6e7
|
200% | Integers |
|
Proofs | Through a series of logical steps, proofs build from agreed-upon assumptions to reach a new conclusion. If x and y are even, x + y is even. Let x = 2a and y = 2b. 2a + 2b = 2(a + b) x + y is a multiple of 2, and is thus even. |
2e12
|
500,000% | Building Blocks Quadratic Formula |
|
Complex Numbers | Complex numbers contain both real and imaginary parts, expressed in the form a+ib. Despite being "imaginary", complex numbers are used in the real world. For example, they are required for compressing MP3s. |
1e17
|
1,000,000% | i |
|
Irrationals | A number is irrational if it cannot be expressed as a fraction. These numbers have decimals that go on forever and don't repeat. The square root of 2 is an irrational number equal to 1.41421356... |
8e17
|
500% | Rationals Complex Numbers |
|
Proof by Contradiction | Sometimes it's easier to prove that the opposite of something is false. One can prove there are infinite primes by assuming there aren't, then showing that if you multiply all primes and add 1, the result would be also be prime. |
2e12
|
2.5e8% | Primes |
|
Proof by Induction | The trick of an inductive proof is showing that if something is true for n, it must be true for n+1. If it can be shown that a proof is true for n=1, and it is true for n+1, it must be true for 2, and thus also 3, 4, 5, ... |
8e26
|
50,000% | Discrete Math Linear Algebra |
|
The Most Beautiful Equation | Euler's Identity unites mathematics' most fundamental concepts in a simple relationship. It elegantly represents complexity in its smallest possible form. ei𝜋+1 = 0 contains addition, multiplication, exponents, pi, e, and i. |
1e19
|
100% | Complex Plane |
Arithmetic Efficiency[]
Arithmetic has 6 upgrades, increasing the generator efficiency with a total x4.63789e23 multiplier.
| Icon | Name | Description | Cost | Efficiency | Requires |
|---|---|---|---|---|---|
|
Addition | Numbers are counted together through addition (+). Subtraction (-) does the opposite, removing a number from another one. Add 3 pears to 2 pears to get 5 pears. Subtract 2 pears from 8 pears for 6 pears. |
8,000
|
400% | Arithmetic |
|
Fibonacci Sequence | The Fibonacci Sequence is a series in which each number is the sum of the previous two numbers. The relationship between these numbers becomes the golden ratio, found in flowers and seashells. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34… |
40,000
|
112.358% | Addition |
|
Multiplication | Operations build on each other. Multiplication (*) gets its name from performing addition multiple times. Division (/) does the opposite. 4 • 3 is 4 + 4 + 4, which is 12. 12 divided into 3 equal parts is 4 + 4 + 4. 12 / 3 = 4. |
1.5e7
|
2,000% | Addition |
|
Factorials | Written as (!), factorials are instructions to "multiply every counting number from here down to 1." One use of factorials is to calculate probability. So 4! is 4•3•2•1. Factorials can grow quickly (15! is 1,307,674,368,000). |
1.5e8
|
100% | Multiplication |
|
Exponentiation | Just like multiplication adds repeatedly, exponentiation (aᵇ) multiplies repeatedly. A number's square root (√) is a value that is multiplied by itself to get the original number. 4³ is 4 • 4 • 4. 2² is 4, the (√4) is 2. |
8e9
|
2,500% | Multiplication Equations |
|
Tetration | Just as multiplication is repeated addition and exponentiation is repeated multiplication, tetration is repeated exponentiation. Tetration grows rapidly: ¹ 4 = 4¹ (1 digit) ² 4 = 4⁴ (3 digits) ³ 4 = 4256 (154 digits) |
5e29
|
4e21% | Exponentiation Game Theory |
Algebra Efficiency[]
Algebra has 8 upgrades, increasing the generator efficiency with a total x9.25385e14 multiplier.
| Icon | Name | Description | Cost | Efficiency | Requires |
|---|---|---|---|---|---|
|
Equations | An equation is a statement that declares each side of an equals sign (=) has the same value. In 2 + 3 = 5, 2 + 3 has the same value as 5, so both expressions are equal. |
2e9
|
300% | Algebra |
|
Variables | Algebra involves solving for unknown values, often represented with symbols such as x, y, a, or 🍌. If you buy 3 pears and now have 7, how many did you start with? The variable x can represent this unknown: x + 3 = 7. |
3e9
|
600% | Equations |
|
Quadratic Formula | Equations that can be written as ax²+bx+c=0 are "quadratic". The quadratic formula solves for x by plugging in a, b, and c. The ± in the formula means there are usually two answers, due to the power of two in the equation. |
2e11
|
400% | Equations Exponentiation |
|
Preserving Equality | An algebraic proof can be solved by manipulating the elements of an equation. When doing this, if an operation is done on one side, the same operation must be done on the other one to preserve equality. 4x=4 4x/4 = 4/4 x=1 |
6e12
|
1,000% | Equations Proofs |
|
Proof that 0.999...=1 | Let x = 0.999... 10•(x) = 10•(0.999...) 10x = 9.999... (10x) - x = (9.999...) - x (remember x = 0.999...) 9x = 9.999... - 0.999... 9x = 9 x = 1 |
1e13
|
99.9% | Preserving Equality |
|
Pascal's Triangle | There are many odd properties of Pascal's Triangle, where each number is the sum of the two above it. Each row adds to a power of 2, the Fibonacci sequence is found in diagonals, and it can count paths a rook can move in chess. |
3e13
|
100% | Algebra Preserving Equality |
|
i | Known as the "imaginary unit", i is √-1. It's imaginary because it is impossible to multiply a real number by itself to get a negative number. Before developing i, many equations were considered unsolvable. |
2e16
|
50,000% | Rationals Trigonmetric Functions |
|
Linear Algebra | A matrix is a set of numbers represented in rows and columns. Linear algebra uses matrices to work with systems of linear equations. Machine learning algorithms are built on linear equations, using them to process inputs. |
1e15
|
3e10% | Equations Discrete Math Infinity |
Applied Math Efficiency[]
Applied Math has 9 upgrades, increasing the generator efficiency with a total x8.52655e16 multiplier.
| Icon | Name | Description | Cost | Efficiency | Requires |
|---|---|---|---|---|---|
|
Compounding Interest | Interest "compounds" over time because the interest you earn now stacks on top of the interest you previously earned. Holding $100 at a rate of 8% for 25 years will end with $685. At 50 years, you would have $4,690. |
2.5e10
|
100% | Equations |
|
Math in Cells | The cost of any generator in Cell to Singularity is determined by using exponential growth. The Applied Math generator has a base cost of 5B and grows 15% per purchase. If you have 30, the next will cost 5B•(1.15)30=331.06B. |
8e11
|
300% | Equations Exponentiation |
|
Voronoi Pattern | Imagine a group of droplets hitting water at the same time. Each ripple will travel outward until hitting another ripple, forming a Voronoi pattern. Voronoi patterns are found throughout nature, such as on a giraffe's spots. |
1.5e17
|
7,500,000% | Four-Color Theorem |
|
Predator-Prey Model | Differential equations can be used to measure rates of change over time. The predator-prey model, when graphed, shows that as prey die out, so do predators, allowing prey populations to grow. This results in a cyclical pattern. |
5e19
|
10,000% | Derivatives |
|
Integrals | The inverse of a derivative, an integral can be used to solve for the area underneath a curve. With a curve of an object's speed, the integral would be the total distance traveled (speed over time). |
1e20
|
250% | Derivatives |
|
Statistics | A branch of applied math, statistics focuses on collecting and analyzing data. If a new medication is being tested among several groups, data can be used to compare results and determine the drug's effectiveness. |
5e20
|
100% | Applied Math Compounding Interest Predator-Prey Model |
|
Normal Distribution | A specific type of bell-shaped curve, a normal distribution can help model the distribution of data points. Naturally occurring data, such as the heights of randomly selected males, often fall into a normal bell curve. |
1e21
|
100% | Statistics |
|
Cryptography | In cybersecurity, information and data can be encrypted using large prime numbers. Take p • q = n. If p and q are both large primes, their values can be difficult to determine, even for anyone who knows the value of n. |
5e22
|
20,000% | Primes |
|
Arrow's Impossibility | Kenneth Arrow proves that rational behavior is impossible in votes where voters choose a single option. For pizza, a choice of ham, olives, or onions could split 6 vegetarians and let 4 carnivores win with ham. |
1e29
|
5e7% | Discrete Math Proof by Induction |
Geometry Efficiency[]
Geometry has 8 upgrades, increasing the generator efficiency with a total x2.24393e11 multiplier.
| Icon | Name | Description | Cost | Efficiency | Requires |
|---|---|---|---|---|---|
|
Area | The space bound by a 2d shape is its area. Areas grow with the square of the dimensions and thus increase quicker than their edges. A 30 cm pizza is over twice as large as a 20 cm pizza. |
1e14
|
100% | Geometry |
|
Pythagorean Theorem | A right triangle has one 90-degree angle, with 2 short sides (a and b), and a long side (c) opposite the 90-degree angle. The Pythagorean Theorem states a²+b²=c². A common right triangle is the 3-4-5 triangle, and 3²+4²=5². |
2.5e14
|
345% | Geometry |
|
Trigonometry | A subset of geometry, trigonometry focuses on the relationship of angles in 2D and 3D objects. Trigonometry is developed by Hellenistic Greeks to study the movement of celestial bodies. |
7.5e14
|
100% | Geometry |
|
Trigonmetric Functions | The three basic trig functions are sine (sin), cosine (cos), and tangent (tan). Sin is opposite side length / hypotenuse length. We know sin 30-degrees is always 1/2, so if the opposite side is 10 the hypotenuse must be 20. |
2e15
|
100% | Trigonometry |
|
Fractals | Geometric shapes that usually appear similar at various scales, fractals cannot be classified into a single dimension. Formed from a 1D line, the Mandelbrot fractal is so intensely "wiggly," it takes up space as a 2D object." |
7e15
|
100% | Pascal's Triangle Geometry Trigonometry |
|
Pi | Probably the most famous irrational number, pi (𝜋) is 3.14159..., the distance around a circle divided by its width. While the number goes on forever, computers have been able to calculate over 105 trillion digits of pi. |
2e18
|
31,415.9% | Geometry Irrationals |
|
Non-Euclidean Geometry | In basic geometry, parallel lines cannot intersect. Non-Euclidean geometry examines what happens when parallel lines can intersect. On the planet Earth, longitude lines are parallel, but they intersect at the poles. |
8e25
|
2e8% | Geometry Countable Infinity |
|
Mobius Strip | A two-dimensional anomaly, the Mobius Strip is a shape with only one continuous side. A Mobius Strip can be created in real life by half-twisting a rectangular strip of paper and connecting one end to the back of the other. |
2.5e26
|
400% | Geometry Linear Algebra |
Marvels and Mysteries Efficiency[]
Marvels and Mysteries has 7 upgrades, increasing the generator efficiency with a total x8.95528e11 multiplier.
| Icon | Name | Description | Cost | Efficiency | Requires |
|---|---|---|---|---|---|
|
Four-Color Theorem | It is possible to color any map so that no adjacent regions share a color. Long achievable with 5 colors, it takes 350 years to prove possible for 4. A mathematical curiosity, it has little relevance to real map-making. |
1,000
|
400% | Geometry Marvels and Mysteries |
|
Irrationalᴵʳʳᵃᵗᶦᵒⁿᵃˡ | An irrational number raised to an irrational exponent can be rational. One proof for this uses √2. Oddly, it doesn't identify specific solutions, but still proves that such numbers do exist. |
1e18
|
50,000% | Rationals Irrationals |
|
Fermat's Last Theorem | Consider the equation aⁿ+bⁿ=cⁿ. Fermat theorizes that for values of n larger than 2, there are no whole-number solutions for a, b, and c. Fermat claims he has a proof but never writes it down. It isn't proven for 350 years. |
8e21
|
50,000% | Pythagorean Theorem Number Theory |
|
Birthday Paradox | What is the chance two people in a group share a birthdate? If there are 23 people, it is over 50%. With just two people, chances are very small, but as each new person gets compared to all the others, the odds rise quickly. |
8e23
|
230% | Probability |
|
Twin Prime Conjecture | Twin primes are prime numbers separated by 2 (like {5, 7} or {8087, 8089}). It is believed there are infinitely-many twin primes. We have found over 800 trillion twin primes up to 1018, but have yet to prove there are infinite. |
5e13
|
35,700% | Primes Infinity |
|
Goldbach's Conjecture | One of math's simplest open questions, Goldbach's Conjecture suggests that every even number greater than 2 can be the sum of two primes. This is true for all numbers up to 1018, but has yet to be proven to infinity. |
8e30
|
15,000% | Number Theory Primes Arrow's Impossibility |
|
Gödel's Incompleteness | Gödel shows there are always statements that cannot be proven true or false in mathematics. The Continuum Hypothesis suggests there is a size of infinity between countable and uncountable. This cannot be proven or disproven. |
5e17
|
300% | Goldbach's Conjecture |
Calculus Efficiency[]
Calculus has 7 upgrades, increasing the generator efficiency with a total x9.10931e9 multiplier.
| Icon | Name | Description | Cost | Efficiency | Requires |
|---|---|---|---|---|---|
|
Limits | What happens as a variable approaches a specific value? Limits provide solutions for these problems. By looking at limits as x gets closer to 0 from positive and negative sides, calculus shows what happens to 1/x at x=0. |
3e8
|
1,500% | Calculus |
|
Derivatives | A derivative measures the rate of change at an exact moment in time, represented as a slope on a graph. If we had a graph of a ball's position over time, the derivative would tell us the rate of change, or speed, over time. |
2e9
|
100% | Limits |
|
e | Euler's number, e, has a unique property - the rate of change of eˣ at any point x is always eˣ. This makes it a key point of stability in much of calculus. Euler's number is irrational, equal to 2.7182818... |
6e9
|
271.828% | Irrationals Limits |
|
Infinity | The list of numbers continues forever, a concept called infinity (∞). 1 / ∞ cannot be solved mathematically, but it can be said that as x approaches infinity, the limit of 1 / x approaches 0. |
8e24
|
10,000% | Calculus Probability |
|
Countable Infinity | If each member of an infinite set can be assigned a positive integer, it is a countable infinite set. All countable infinite sets are considered the same size. Integers and rational numbers are both countably infinite sets. |
1.5e13
|
400% | Infinity |
|
Uncountable Infinity | Sets are uncountably-infinite if it is impossible to line elements up 1:1 with a countable set. The set of irrational numbers is uncountable, meaning there are infinitely many more irrationals than there are countable rationals. |
3e27
|
10,000% | Countable Infinity Mobius Strip |
|
Complex Plane | Complex functions are drawn on the complex plane, where real numbers are graphed on the x-axis and imaginary numbers are on the y-axis. On this plane, sin and cos can be converted into eix, which is often easier to manipulate. |
1.5e18
|
150,000% | Trigonmetric Functions Goldbach's Conjecture |
Discrete Math Efficiency[]
Discrete Math has 4 upgrades, increasing the generator efficiency with a total x2.09553e6 multiplier.
| Icon | Name | Description | Cost | Efficiency | Requires |
|---|---|---|---|---|---|
|
Number Theory | A branch of discrete math, number theory limits problems to only integer solutions. For x³+y³+z³=33, the smallest integer solution is: x = 8,866,128,975,287,528 y = -8,778,405,442,862,239 z = –2,736,111,468,807,040 |
2e21
|
1,500% | Discrete Math |
|
Primes | A positive integer divisible only by itself and the number 1 is a prime. Numbers that aren't prime are called composite. 6 is divisible by 6, 3, 2, and 1. 7 can only be divided by 7 and 1. 6 is composite. 7 is prime |
5e11
|
235.711% | Number Theory |
|
Probability | The chances of outcomes occurring from a particular event is referred to as probability. There are 36 possible results of rolling two 6-sided dice. 5 of those results add up to 8, so the probability of getting an 8 is 5/36. |
3e23
|
1,200% | Statistics Discrete Math Primes |
|
Game Theory | Thought experiments mathematically explore why people make certain choices when interacting with each other. The prisoner's dilemma sets up an individual vs group dynamic, with each participant incentivized to betray the others. |
2e16
|
300,000% | Discrete Math Proof by Induction |
Tech Tree[]
|
Trivia[]
- This is the first exploration not to focus on Science or Social Studies.
- The event was originally named Crunch the Numbers, but got renamed when it was still in beta.
- The upgrades e and Pi have their starting digits as the efficiencies for the generators Geometry and Calculus respectively.
- In the beta, Semblance's opening quote for this exploration was "math, Math, MATH".
 Explorations and Events
| |
|---|---|
| Season 1 |  James Webb Telescope •  Fungi •  Philosophy •  Mass Extinction •  Money •  Pollination
|
| Season 2 |  Deep Sea Life •  Tea •  Music •  Human Body •  Visual Art •  Outbreaks •  Cats •  Rocks •  Cryptids •  Mathematics •  Artificial Intelligence •  Cheese
|
| Special Events |  A Dodo Ghost Hunt •  Holiday Advent Calendar
|