144 Circles, Hexagons, Triangles, Squares, and Other Unexpected Shapes in Math and Nature

Since the days of ancient Greek philosophers like Plato and Pythagoras, mathematicians and scientists have been captivated by the beauty and order of geometric shapes. From the perfect circle to the humble triangle, from the elegant hexagon to the playful pentagon, these shapes have played a fundamental role in our understanding of the world around us.

In this article, we will explore 144 examples of circles, hexagons, triangles, squares, and other shapes found in both the natural world and the realm of mathematics. We will see how these shapes manifest themselves in everything from the honeycomb of a beehive to the intricate patterns of a snowflake, from the spirals of a seashell to the symmetries of a flower. Along the way, we will learn about the mathematical properties of these shapes and their significance in various fields of science and engineering.

Beyond the Square Crochet Motifs: 144 circles, hexagons, triangles, squares, and other unexpected shapes

by Edie Eckman

4.8 out of 5

Language	:	English
File size	:	59537 KB
Text-to-Speech	:	Enabled
Screen Reader	:	Supported
Enhanced typesetting	:	Enabled
X-Ray	:	Enabled
Print length	:	658 pages
Lending	:	Enabled

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Circles

Circles are one of the most fundamental shapes in mathematics. They are defined by their constant radius, which is the distance from the center of the circle to any point on the circle. Circles have a number of interesting properties, including:

• Circles are symmetric about their center.
• The circumference of a circle is equal to $\backslash pi$

  times the diameter.

• The area of a circle is equal to $\backslash pi$

  times the radius squared.

Circles are found in a wide variety of natural and mathematical contexts. They are the shape of the sun, the moon, and the planets. They are also found in the petals of flowers, the bubbles of soap, and the ripples of water. In mathematics, circles are used to define a variety of other shapes, such as spheres, cones, and cylinders.

Hexagons

Hexagons are polygons with six sides. They are regular hexagons when all six sides are equal in length. Regular hexagons have a number of interesting properties, including:

• Regular hexagons are symmetric about their center.
• The interior angles of a regular hexagon are all equal to 120 degrees.
• The area of a regular hexagon is equal to $\backslash frac\{3\backslash sqrt\{3\}\}\{2\}$

  times the square of the side length.

Hexagons are found in a variety of natural and mathematical contexts. They are the shape of the honeycomb cells of bees and wasps. They are also found in the crystalline structure of graphite and in the patterns of snowflakes. In mathematics, hexagons are used to define a variety of other shapes, such as prisms and pyramids.

Triangles

Triangles are polygons with three sides. They are the simplest of all polygons, and they have a number of interesting properties, including:

• Triangles are symmetric about their centroid, which is the point where the medians of the triangle intersect.
• The sum of the interior angles of a triangle is always 180 degrees.
• The area of a triangle is equal to $\backslash frac\{1\}\{2\}$

  times the base times the height.

Triangles are found in a wide variety of natural and mathematical contexts. They are the shape of mountain peaks, the sails of boats, and the wings of birds. They are also used in the construction of bridges, buildings, and other structures. In mathematics, triangles are used to define a variety of other shapes, such as tetrahedra, pyramids, and prisms.

Squares

Squares are rectangles with four equal sides. They are regular quadrilaterals, meaning that they have four equal sides and four right angles. Squares have a number of interesting properties, including:

• Squares are symmetric about their center.
• The diagonals of a square are perpendicular to each other and bisect each other.
• The area of a square is equal to the square of the side length.

Squares are found in a variety of natural and mathematical contexts. They are the shape of the faces of a cube, the tiles of a chessboard, and the pixels of a computer screen. In mathematics, squares are used to define a variety of other shapes, such as cubes, pyramids, and prisms.

Other Shapes

In addition to circles, hexagons, triangles, and squares, there are a variety of other shapes that can be found in nature and mathematics. These shapes include:

• Pentagons are polygons with five sides.
• Octagons are polygons with eight sides.
• Dodecagons are polygons with twelve sides.
• Ellipses are curves that are similar to circles but have two different radii.
• Parabolas are curves that are formed by the intersection of a cone and a plane.
• Hyperbolas are curves that are formed by the intersection of two cones.

These are just a few examples of the many different shapes that can be found in nature and mathematics. Each shape has its own unique properties and applications. By studying these shapes, we can learn more about the world around us and the underlying principles that govern it.

Circles, hexagons, triangles, squares, and other shapes are all around us, in both the natural world and the realm of mathematics. These shapes have a variety of interesting properties and applications, and they play a fundamental role in our understanding of the world around us. By studying these shapes, we can learn more about the universe and our place in it.

Beyond the Square Crochet Motifs: 144 circles, hexagons, triangles, squares, and other unexpected shapes

by Edie Eckman

4.8 out of 5

Language	:	English
File size	:	59537 KB
Text-to-Speech	:	Enabled
Screen Reader	:	Supported
Enhanced typesetting	:	Enabled
X-Ray	:	Enabled
Print length	:	658 pages
Lending	:	Enabled

FREE