9.4 Symmetry in two and three dimensions
3 min read•july 22, 2024
Symmetry in geometry is all about balance and repetition. In 2D, we look at line and , where shapes can be split into identical halves or rotated to match their original position. It's like finding hidden patterns in everyday objects.
3D symmetry takes things up a notch. We explore plane symmetry, where 3D figures can be divided into mirror images. From cubes to spheres, different shapes have unique symmetry properties, revealing the beauty and order in three-dimensional space.
Symmetry in Two Dimensions
Line and rotational symmetry
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- occurs when a figure can be divided into two identical halves by a line called the line of symmetry
- Folding the figure along the line of symmetry results in the two halves coinciding perfectly (equilateral triangle, )
- Some figures have multiple lines of symmetry (rectangle has 2, has infinite)
- Rotational symmetry happens when a figure can be rotated less than 360° about its center point and still appear identical to its original orientation
- The smallest angle that maps the figure onto itself is the
- Figures with rotational symmetry include regular polygons (equilateral triangle, square, regular pentagon) and circles
Symmetry order in figures
- Number of lines of symmetry varies among different shapes
- Equilateral triangle has 3 lines of symmetry passing through each vertex and midpoint of opposite side
- Square features 4 lines of symmetry connecting midpoints of opposite sides and diagonals
- Rectangle possesses 2 lines of symmetry through midpoints of opposite sides
- Circle has infinite lines of symmetry as any diameter can divide it into identical halves
- Regular pentagon exhibits 5 lines of symmetry connecting each vertex to midpoint of opposite side
- Order of rotational symmetry represents the number of distinct orientations in which the figure appears the same during a complete 360° rotation
- Equilateral triangle: order 3 (rotations of 120°, 240°, 360° produce identical appearance)
- Square: order 4 (rotations of 90°, 180°, 270°, 360° yield the same orientation)
- Rectangle: order 2 (rotations of 180° and 360° map the figure onto itself)
- Circle: infinite order (any angle of rotation results in the same appearance)
- Regular pentagon: order 5 (rotations of 72°, 144°, 216°, 288°, 360° produce identical orientations)
Symmetry in Three Dimensions
Plane symmetry in 3D
- Plane symmetry in three-dimensional figures occurs when a plane can divide the figure into two identical halves
- The dividing plane is known as the
- Examples of 3D figures with plane symmetry include cubes, regular tetrahedra, octahedra, spheres, and dodecahedra
- across the plane of symmetry results in the two halves coinciding perfectly
Types of 3D symmetry
- Cube symmetry:
- 9 planes of symmetry (3 through opposite faces, 6 through opposite edges)
- Rotational symmetry: order 3 about face diagonals, order 4 about face-to-face axes, order 2 about edge-to-edge axes
- Regular tetrahedron symmetry:
- 6 planes of symmetry passing through opposite edges
- Rotational symmetry: order 3 about vertex-to-face axes, order 2 about edge-to-edge axes
- Regular octahedron symmetry:
- 9 planes of symmetry (3 through opposite vertices, 6 through opposite edges)
- Rotational symmetry: order 4 about vertex-to-vertex axes, order 3 about face diagonals, order 2 about edge-to-edge axes
- Sphere symmetry:
- Infinite planes of symmetry passing through the center
- Rotational symmetry of infinite order about any axis through the center
- Regular dodecahedron symmetry:
- 15 planes of symmetry (3 through opposite vertices, 6 through opposite faces, 6 through opposite edges)
- Rotational symmetry: order 5 about vertex-to-vertex axes, order 3 about face-to-face axes, order 2 about edge-to-edge axes
Key Terms to Review (18)
Angle of rotation: The angle of rotation refers to the degree measure of the turn about a specific point, typically the center of a geometric figure. This angle is crucial for understanding rotational symmetry in both two and three dimensions, as it determines how far a shape can be rotated before it aligns with its original position. Recognizing the angle of rotation helps in identifying symmetrical properties and establishing how figures relate to one another in space.
Axis of Symmetry: The axis of symmetry is a line that divides a figure into two mirror-image halves, ensuring that for any point on one side of the line, there is a corresponding point on the other side at an equal distance from the axis. This concept is fundamental in understanding the symmetrical properties of shapes, especially in relation to graphs and geometric figures, as well as in analyzing the behavior of quadratic functions and conic sections.
Bilateral symmetry: Bilateral symmetry is a type of symmetry where an object can be divided into two identical halves along a single plane, creating mirror images on either side. This concept is significant in understanding shapes and forms in both two and three dimensions, as it highlights how certain objects exhibit uniformity and balance when viewed from different angles. Recognizing bilateral symmetry can be essential in various fields, including biology, art, and architecture, where this symmetry often signifies harmony and proportionality.
Circle: A circle is a two-dimensional geometric shape consisting of all points that are equidistant from a fixed center point. This distance is known as the radius, and the line segment connecting the center to any point on the circle is also a radius. Circles exhibit unique properties, such as constant curvature and symmetry, making them fundamental in both geometry and various applications in mathematics and science.
Congruence: Congruence refers to the property of geometric figures being identical in shape and size, allowing them to be superimposed onto one another without any gaps or overlaps. This concept is essential when comparing figures and helps in understanding their relationships, particularly in the study of angles, sides, and various transformations. Congruent figures maintain their properties through transformations like translations, reflections, and rotations, which are fundamental in geometry.
Fractal symmetry: Fractal symmetry refers to the self-similar patterns that repeat at different scales, creating a complex structure that looks similar regardless of how closely you examine it. This concept is deeply connected to geometry and nature, where fractals appear in various forms, such as snowflakes, coastlines, and even in the branching patterns of trees. Fractal symmetry is a fascinating intersection of mathematics and art, illustrating how simple rules can lead to intricate designs.
Line symmetry: Line symmetry, also known as reflectional symmetry, occurs when a figure can be divided into two identical halves that are mirror images of each other along a line. This concept is crucial in understanding how shapes can be transformed through reflections and how they relate to other symmetrical properties in both two and three dimensions.
Order of Rotation: The order of rotation refers to the number of times a shape can be rotated around a central point and still look the same within a full rotation of 360 degrees. This concept is closely linked to symmetry, as it helps identify the degree of rotational symmetry a shape possesses, whether in two-dimensional figures like polygons or three-dimensional objects like spheres and cubes.
Plane of symmetry: A plane of symmetry is an imaginary plane that divides a three-dimensional object into two mirror-image halves. This concept is essential for understanding symmetry in shapes and forms, as it helps identify how objects can be reflected across that plane without altering their appearance. In two dimensions, a line can serve as a plane of symmetry, demonstrating how figures can exhibit balanced proportions around a central line.
Reflection: Reflection is a transformation that flips a figure over a line, creating a mirror image of the original shape. This concept is crucial for understanding how shapes can be manipulated in space and relates to various geometrical operations like translations and rotations, symmetry in different dimensions, applications of vectors, and the composition of transformations.
Regular polygon: A regular polygon is a geometric figure with all sides and angles equal, creating a symmetrical and uniform shape. This property of equal length sides and equal angles is crucial for understanding various mathematical concepts, such as area calculations and symmetry. Regular polygons include shapes like equilateral triangles, squares, and regular pentagons, and they can be used to form more complex figures.
Rotational Symmetry: Rotational symmetry is a characteristic of a shape that looks the same after being rotated by a certain angle around a central point. This concept is crucial in understanding how shapes and objects can be transformed and how they maintain their identity through various movements and orientations.
Similarity: Similarity refers to the relationship between two shapes or figures that have the same shape but may differ in size. This concept is crucial in geometry as it allows for the comparison of geometric figures through ratios and proportions, enabling us to understand and analyze their properties regardless of their size. When two figures are similar, their corresponding angles are equal, and the lengths of their corresponding sides are proportional.
Square: A square is a special type of quadrilateral that has four equal sides and four right angles. It is a unique shape because it possesses the properties of both a rectangle and a rhombus, making it a versatile figure in geometry. Squares play a significant role in understanding the broader classification of quadrilaterals, parallelograms, symmetry, and special types of quadrilaterals.
Symmetrical design: Symmetrical design refers to a balanced and proportionate arrangement of elements where one side mirrors the other, creating harmony and order. This concept is not only significant in art and architecture but also in mathematics and geometry, as it showcases how shapes and forms can be evenly distributed around a central axis or point, resulting in visually appealing compositions in both two and three dimensions.
Symmetrical patterns: Symmetrical patterns are arrangements that display balance and proportion, where one half mirrors or corresponds to the other half. This concept is essential in geometry, as it reveals the inherent structure in shapes and forms, making them aesthetically pleasing and mathematically significant. Symmetry can manifest in both two and three dimensions, creating a variety of visual effects and structural integrity.
Symmetry Group: A symmetry group is a mathematical concept that captures the symmetries of an object by describing the set of transformations that leave the object unchanged. This includes operations such as rotations, reflections, and translations, all of which can be combined in various ways to demonstrate how an object can be manipulated while maintaining its original form. The study of symmetry groups helps in understanding the geometric properties of shapes and forms in both two and three dimensions.
Translation: Translation is a type of transformation that moves every point of a shape or object a certain distance in a specified direction, without changing its size, shape, or orientation. This concept is crucial for understanding how figures can be manipulated on a coordinate plane, and it connects to reflections, rotations, and various applications in geometry and vectors.