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Symmetry

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
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- Decision Maths
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- Area of a Kite
- Composition
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- Coordinate Systems
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- Equation of Circles
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- Figures
- Fundamentals of Geometry
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- Glide Reflections
- HL ASA and AAS
- Identity Map
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- Isometry
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- Law of Cosines
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- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
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- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
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- Special quadrilaterals
- Squares
- Surface Area of Cone
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- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
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- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
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- Volume of Cone
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- Volume of Solid
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- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
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- Constant Acceleration
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- Kinematics
- Newton's First Law
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- Projectiles
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- Variable Acceleration
- Probability and Statistics
- Bar Graphs
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- Charts and Diagrams
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- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
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- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
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- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
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- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
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- Arithmetic Sequences
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- Combination of Functions
- Combinatorics
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- Completing the Square
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- Complex Numbers
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- Composition of Functions
- Compound Interest
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- Construction and Loci
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- Coordinate Geometry
- Coordinates in Four Quadrants
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- Deductive Reasoning
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- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
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- Differentiation of Hyperbolic Functions
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- Distance from a Point to a Line
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- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
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- Fractions
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- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
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- Lowest Common Multiple
- Math formula
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- Matrix Addition and Subtraction
- Matrix Determinant
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- Metric and Imperial Units
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- Modulus Functions
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- Multiples of Pi
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- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
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- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
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- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
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- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
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- SSS Theorem
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- Segment of a Circle
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- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
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- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
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- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
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- Special Products
- Standard Form
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- Sum and Difference of Angles Formulas
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- Surds
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- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
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- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
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- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
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- Verifying Trigonometric Identities
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- Writing Linear Equations
- Statistics
- Bias in Experiments
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- Bivariate Data
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- Categorical Variables
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- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
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- Type I Error
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- Types of Data in Statistics
- Venn Diagrams

Symmetry can be seen in nature all around us, such as when a mountain range is reflected on the water of a lake.

In this article, we will cover the concept of symmetry, differentiating between different types. We will also discuss how the principles of symmetry can be applied to figures on graphs.

Most likely, you have already encountered the concept of symmetry outside of math. For example, when you think of symmetry, you may think of a mirror image. Indeed, a mirror is an excellent example of symmetry in the real world. Mirrors showcase a type of symmetry called **reflective symmetry**, in which one half of the object is a mirror or **reflection** of the other. Reflective symmetry is only one type of symmetry, however. The definition of symmetry in mathematics is more precise than the common usage of symmetry in everyday life. A formal definition is as follows:

In mathematics, symmetry is when two or more objects are identical after a transformation has been preformed, such as a flip, a slide, or a turn. The mathematical object could be a shape within a plane or a line on a graph, for example.

In this section, we will cover four different types of symmetry:

- Translational symmetry
- Rotational symmetry
- Reflective symmetry
- Glide symmetry

Translational symmetry occurs when an object has undergone a shift or translation (i.e., it has had its location changed), but there have been no other transformations performed on it. In other words, translational symmetry only involves the object moving position, such as up or down as well as left or right. In translational symmetry, the object will remain the same size and shape, and it won't rotate in any way.

Let's take the figure below as an example and see how translational symmetry looks:

Now, let's translate the figure up and to the left:

You can see that the figure has not changed size or shape, and it hasn't been rotated. It has simply moved position. For these reasons, it is an example of translational symmetry.

Rotational symmetry applies if a shape can be partially rotated about its center and still look the same. In other words, for any object to be rotationally symmetrical, it must have at least two positions within the full turn of rotation (360 degrees) where it looks identical.

To describe rotational symmetry, we may refer to the **order of rotational symmetry**, which describes the number of times that the object is identical to its original position within the full rotation of 360 degrees. For example, an equilateral triangle has a rotational symmetry of order 3 because it can be partially rotated 3 times and still appear identical. To find out the order of rotational symmetry of a shape, you can use the following equation:

As you can see in the diagram above, point A is moving with each rotation, but the triangle itself looks exactly the same. In the case of this triangle, we say that it has rotational symmetry of order 3 because there are 3 positions that the triangle can be rotated to where it will have symmetry. Let's check this by looking at the formula to find the order of rotational symmetry:

Point symmetry occurs when there is a common point of reflection for every point on a shape. That common point is called a point of symmetry. Note that the reflection for each point is in the opposite direction, so that it looks the same from the top as it does from the bottom. An example of point symmetry is the letter H.

An important thing to note is that point symmetry is the same as a shape having rotational symmetry of order 2, or in other words, the object looks identical after you have rotated it 180 degrees about its center.

Reflective symmetry is a type of symmetry in which one half of the object reflects the other half. Reflective symmetry is known by several different names, including line symmetry and mirror symmetry. All three terms have the same meaning: one half of the object is identical to the other.

To check if an object exhibits reflective symmetry, imagine folding it in half along the line of symmetry (an imaginary line that divides the two reflective halves). If both halves match, then the shape has reflective symmetry along the line that it was folded over.

In this example, the original square is labelled CDEF, and the line of symmetry is the y-axis. As you can see, the reflected image has the same y-value coordinates but has negative x-values.

The** line of symmetry** is a line that can be drawn on an object whereby both sides will reflect one another, cutting the object in half. For example, if you place a line of symmetry down the middle of a square, both sides will be the same.

Different shapes can have varying amounts of lines of symmetry:

- A square has 4 lines of symmetry
- A equilateral triangle has 3 lines of symmetry
- A trapezium doesn't have any lines of symmetry

Symmetry isn't reserved for shapes and patterns: it can also be seen in graphs. When graphs have symmetry, it can be a useful property because it allows us to predict and better understand the symmetric portions. Generally, graphs are symmetric about an **axis of symmetry.**

Like the line of symmetry, the axis of symmetry is a straight line over which an object is reflected in order to obtain two equal and mirror parts. If a graph was symmetrical for both negative and positive values of x, then it would have an axis of symmetry along the y-axis. As a straight line, we can describe the axis of symmetry with the equation of a line: . Let's take a look at an example that demonstrates the axis of symmetry.

Let's consider the graph below and determine if there is any symmetry.

In this graph of a parabola, there is a vertical axis of symmetry, as the equation of the parabola is. We can see that the lowest point of the graph is at y = -2. This is also the point on the graph where the parabola curve crosses over the y-axis, and, therefore, the location of the axis of symmetry must be the y-axis, or x = 0.

Glide reflection is a combination of two different transformations, a translation and a reflection, although not necessarily in that order. The translation is always parallel to the line of reflection in a glide reflection. If the reflected image moves further away or closer to the line of reflection, then it is not a case of glide reflection.

Glide reflections can create symmetry. When two glide reflections happen, the original image returns back to the same position, creating a reflection and a line of symmetry between the two images.

A real-life example of glide symmetry is footprints in the sand. When we compare the left footprint with the right footprint in the image below, we see that the left footprint can be reflected over a line of reflection and translated up along that line to obtain the print of the right foot.

- Symmetry occurs when two or more objects are identical after a transformation has been performed.
- There are 4 main types of symmetry: translational symmetry, rotational symmetry, reflective symmetry, and glide symmetry.
- Translational symmetry is where an object has been moved (translated) but still looks the same.
- Rotational symmetry is where an object has been rotated around its center and looks the same in more than one position.
- Reflective symmetry is where an object is reflected over an axis or a line, resulting in one half looking exactly the same as the other.
- Glide symmetry combines two steps of transformations, a reflection and then a translation, although not necessarily in that order.

** line of symmetry** is a line that can be drawn on an object whereby both sides will reflect one another, cutting the object in half. For example, if you place a line of symmetry down the middle of a square, both sides will be the same.

More about Symmetry

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