This diagram is based off of the book in this footnote. This proof turns the and/ or to exclusively to or when added to the proof that any closed surface is homeomorphic to a sphere, and/ or to a connected sum of tori, and/ or to a connected sum of projective planes (also not yet proven on this page). Thus, the connected sum of any number of tori and real projective planes can be reduced solely to a sum of real projective planes. It is crucial to note, however, that you cannot simply subtract a RPP from both sides, because: Thus, as stated in the pictures (# means: the connected sum): Since a Klein bottle is homeomorpic to the connected sum of two Projective Planes (not demonstrated), the connected sum of a Real Projective Plane and a Torus is homeomorphic to that of three Real Projective Planes. Rough diagram of how the connected sum of a Real Projective Plane and a Torus is homeomorphic to that of a Real Projective Plane and a Klein bottle. A hexagonal prism has identical hexagons at each end: triangular prism, square prism (also called a cube), rectangular prism, pentagonal prism, hexagonal prism, heptagonal prism, octagonal prism.This is a preliminary page that needs development. For instance, a rectangular prism has identical rectangles at each end. Named after the two-dimensional shape at the ends. Prism: A three-dimensional figure with identical ends.Pyramids are named after the shape of their base (triangular pyramid, square pyramid, rectangular pyramid). Pyramid: A three-dimensional figure on which the faces are triangular and converge to a single point at the top.Cone: A solid or hollow object that tapers from a circular or roughly circular base to a point.Quadrilaterals (Rhombus, Parallelogram, etc) Rectangle, Rhombus, Square, Parallelogram, Trapezoid and Kite. Sphere: A round body whose surface is at all points equidistant from the center. Plane Geometry is all about shapes on a flat surface (like on an endless piece of paper).Cube: A three-dimensional figure with identical equilateral squares as faces (a cube is also a prism, see square prism below).Octagon: A closed figure with eight sides.
Septagon/Heptagon: A closed figure with seven sides.Hexagon: A closed figure with six sides.Pentagon: A closed figure with five sides.Square: A special type of rectangle that has equilateral (same length) sides and parallel opposite sides.Rhombus: A closed four-sided figure with parallel opposite sides.Rectangle: A two-dimensional, closed, four-sided figure with four right angles.Triangle: A closed figure with three sides.Circle: A two-dimensional shape in which all points on the curved line are equidistant from a center point.Wavy: Having curvy or undulating attributes (e.g., ocean, octopus tentacles).The point at which two sides of a two-dimensional figure or two edges of a three-dimensional figure meet. Vertex/Vertices: Also known as corner/corners.Side: Line segments in geometric figures that compose the exterior of the object.Pointy: An informal word to describe angles of objects.Point: A specific position on a line, plane, or in space.Plane: A flat two-dimensional surface upon which two-dimensional objects (e.g., circle, square) can lie, or the face of a three-dimensional object as it creates a surface (e.g., face of a cube) the dimensions of which are infinite.A parallelogram has two sets of parallel lines. Parallel: Two lines in a two-dimensional space that do not meet (for example, the opposite sides of a square).Location terms: Words and phrases that describe the location of an object in relation to other objects, including words like on, under, on top of, next to, in front of, behind, beneath, underneath, inside, outside, at/on the bottom/top, between, and adjacent to (this is just a sample, there are many more).Equilateral: Sides that are the same length.Face: Surface planes of three-dimensional shapes.Edge: The meeting of two faces on a three-dimensional shape.With young children, the terms large, medium, small, taller, shorter, longer, less than and greater than are all appropriate. Dimensions/Size: Terminology to describe the dimensions of an object or set.Congruent: Identical in size and shape. Other Words from plane Synonyms & Antonyms Choose the Right Synonym Example Sentences.Angle: Two lines that meet to make a corner.Qualities (usually used with geometry) of a figure, including sides, size, angles, etc.
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