Math1089

Geometry, derived from the Greek (γεωμετρία) word Geometron, is the study of the properties and relationships of points, lines, shapes, and spaces. The term geo refers to Earth, while metron means measurement.

The sum of the internal angles of a triangle is equal to 180 degrees. This property is known as the angle sum property of a triangle.

Did you ever notice that the cells inside beehives are hexagonal? Interestingly, each wax cell may vary slightly in size, but they are all alike in their basic shape, which is a hexagon. This shape is a distinctive feature of a honeycomb, which is created by bees to store their larvae, honey, and pollen.

The Seven Bridges of Königsberg is a famous problem named after the city of Königsberg, which was connected to two large islands by seven bridges. The problem, first posed in 1736 by Leonhard Euler, asked whether it was possible to find a walk through the city that would cross each bridge exactly once and end at the starting point. Euler proved that such a walk was impossible, laying the foundation for the modern field of graph theory.

This formula is named after the mathematician Leonhard Euler, who first discovered it in the 18th century. In any convex polyhedron (a three-dimensional solid with flat polygonal faces), the number of its vertices (V), edges (E), and faces (F) are related by the equation:

V ⎼ E + F = 2.

The Exterior Angle Sum Theorem in geometry states that the sum of the exterior angles of a polygon is always equal to 360 degrees. Regardless of whether the polygon is a triangle, quadrilateral, hexagon, or decagon, the sum of its exterior angles remains constant.

Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity that starts at A and reaches B in the shortest time? This classical problem is known as the Brachistochrone problem, which was posed and solved by Bernoulli in 1696. The curve of fastest descent is not a straight line or a polygonal line but a cycloid.

Potato chips come in various shapes and sizes, but Pringles are unique due to their saddle shape, which is a hyperbolic paraboloid. This shape was chosen because it allows for uniform stacking of the chips in the canister, maximizing storage space while minimizing breakage. Furthermore, the saddle shape ensures that each chip is evenly cooked during the manufacturing process, resulting in consistent flavor and texture.

Mathematicians have determined that there are only 17 distinct types of planar symmetry that can occur in a repeating wallpaper pattern.  However, it’s important to note that within each of these 17 groups, there are many possible variations in terms of colour, texture, and other decorative elements.

Typical ceiling fans have three blades, each inclined to the adjacent at an angle of 120°. The 3 blades are positioned at equal angles so that the net force they apply becomes 0 and there isn’t any force remainder in any direction.

There are 43252003274489856000 possible configurations of the Rubik’s Cube. Every Rubik’s Cube configuration can be solved in 20 moves or less. This is known as God’s Number. The 3 × 3 × 3  Rubik’s Cube contains 26 unique miniature cubes, also known as cubies (or cubelets).

1 radian is more than 1 degree. In fact, 1 radian is approximately equal to 57.2958º.

The main supporting structure for overhead electric transmission lines is the towers. Transmission towers are designed to bear the weight of heavy transmission cables and elevate them above the ground. Triangular shapes are often used in structures such as electric towers (known as cross arms) because they are rigid and stable. A triangle’s shape distributes forces evenly, making it stronger and more resistant to bending or breaking compared to a quadrilateral. Additionally, triangular structures use less material to achieve the same level of stability, which makes them more cost-effective to build.

If a ball is projected at an angle of 45 degrees with respect to the horizontal, it will travel the maximum distance possible.

The international paper sizes such as A₃, A₄, and others, are based on the square root of 2. The dimensions of an A₄ size paper are 210 mm × 297 mm, and that of an A₃ size paper is 297 mm × 420 mm. If we assume that the height is greater than the width, then the ratio of the height to the width of all formats is √2 or 1.414. Joining two A₄ size papers will result in one A₃ size paper, two A₃ size papers will give you one A₂ size paper, and so on.

A platonic solid is a 3D shape where each face is the same as a regular polygon and has the same number of faces meeting at each vertex. There are only five Platonic solids and they are tetrahedron, cube, octahedron, dodecahedron, and icosahedron. They were first discovered by the ancient Greek philosopher Plato, who believed that they represented the five elements of the universe: earth, air, fire, water, and ether.

Normally pizza is round, has a square box for delivery but is cut into triangles. Pizzas are mostly circular as it allows for even cooking and heat distribution in a round oven, resulting in a delicious and evenly-cooked pizza. Square pizza boxes are more practical and economical as they fit neatly, stack easily, and take up less space than circular boxes during transport. They are also easier and less expensive to produce. Cutting pizza into triangular slices is practical and convenient, as it ensures equal portions of toppings and crust, maximizes the number of slices, and allows for easy sharing and portion control.