|
Theorem Of The Gnomon
The theorem of the gnomon states that certain parallelograms occurring in a gnomon have areas of equal size. Theorem In a parallelogram ABCD with a point P on the diagonal AC, the parallel to AD through P intersects the side CD in G and the side AB in H. Similarly the parallel to the side AB through P intersects the side AD in I and the side BC in F. Then the theorem of the gnomon states that the parallelograms HBFP and IPGD have equal areas.Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: ''Mit harmonischen Verhältnissen zu Kegelschnitten: Perlen der klassischen Geometrie''. Springer 2016, , pp. 190-191William J. Hazard: ''Generalizations of the Theorem of Pythagoras and Euclid's Theorem of the Gnomon''. The American Mathematical Monthly, volume 36, no. 1 (Jan., 1929), pp. 32–34JSTOR ''Gnomon'' is the name for the L-shaped figure consisting of the two overlapping parallelograms ABFI and AHGD. The parallelograms of equal area HBFP and IPGD are called ''complements'' (o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gnomon Theorem
A gnomon (; ) is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields. History A painted stick dating from 2300 BC that was excavated at the astronomical site of Taosi is the oldest gnomon known in China. The gnomon was widely used in ancient China from the second century BC onward in order to determine the changes in seasons, orientation, and geographical latitude. The ancient Chinese used shadow measurements for creating calendars that are mentioned in several ancient texts. According to the collection of Zhou Chinese poetic anthologies ''Classic of Poetry'', one of the distant ancestors of King Wen of the Zhou dynasty used to measure gnomon shadow lengths to determine the orientation around the 14th century BC. The ancient Greek philosopher Anaximander (610–546 BC) is credited with introducing this Babylonian instrument to the Ancient Greeks. The ancient Greek mathematician and astronomer Oenopides used th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped. The etymology (in Greek παραλληλ-όγραμμον, ''parallēl-ógrammon'', a shape "of parallel lines") reflects the definition. Special cases *Rectangle – A parallelogram with four angles of equal size (right angles). *Rhombus – A parallelogram with four sides of eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gnomon (figure)
In geometry, a gnomon is a plane figure formed by removing a similar parallelogram from a corner of a larger parallelogram; or, more generally, a figure that, added to a given figure, makes a larger figure of the same shape. Building figurate numbers Figurate numbers were a concern of Pythagorean mathematics, and Pythagoras is credited with the notion that these numbers are generated from a ''gnomon'' or basic unit. The gnomon is the piece which needs to be added to a figurate number to transform it to the next bigger one. For example, the gnomon of the square number is the odd number, of the general form 2''n'' + 1, ''n'' = 1, 2, 3, ... . The square of size 8 composed of gnomons looks like this: ~~~~~~~~\begin 8&8&8&8&8&8&8&8\\ 8&7&7&7&7&7&7&7\\ 8&7&6&6&6&6&6&6\\ 8&7&6&5&5&5&5&5\\ 8&7&6&5&4&4&4&4\\ 8&7&6&5&4&3&3&3\\ 8&7&6&5&4&3&2&2\\ 8&7&6&5&4&3&2&1 \end To transform from the ''n-square'' (the square of size ''n'') to the (''n'' + 1)-square, one adjoins 2''n'' + 1 elements: on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A pl ... or planar lamina, while ''surface area'' refers to the area of an open surface or the boundary (mathematics), boundary of a solid geometry, three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a plane curve, curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). The area of a shape can be measured by com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johannes Tropfke
Johannes Tropfke (14 October 1866 – 10 November 1939) was a German mathematician and teacher, who is best remembered for his influential work on the history of mathematics ''Geschichte der Elementarmathematik'', which consists of seven volumes. Life Tropfke was born in Berlin at Marienstraße 14 as the older of two sons of the cabinet maker Franz Tropfke. The house in which Tropfke was born was built by his grandfather Franz Joseph Tropfke around 1830 and is one of the few houses in the area that was not destroyed during World War II. Tropfke grew up in Berlin and after his graduation from the Friedrichs-Gymnasium (high school) in 1884 he attended the university in Berlin to study sciences and mathematics. In 1889 he was awarded a degree to teach math and sciences at Gymnasium (Germany), gymnasiums (high schools). Later he earned a PhD in mathematics from the Martin Luther University of Halle-Wittenberg, University of Halle for a thesis on elliptic integrals (''Zur Darstellung d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gnomon Division
A gnomon (; ) is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields. History A painted stick dating from 2300 BC that was excavated at the astronomical site of Taosi is the oldest gnomon known in China. The gnomon was widely used in ancient China from the second century BC onward in order to determine the changes in seasons, orientation, and geographical latitude. The ancient Chinese used shadow measurements for creating calendars that are mentioned in several ancient texts. According to the collection of Zhou Chinese poetic anthologies '' Classic of Poetry'', one of the distant ancestors of King Wen of the Zhou dynasty used to measure gnomon shadow lengths to determine the orientation around the 14th century BC. The ancient Greek philosopher Anaximander (610–546 BC) is credited with introducing this Babylonian instrument to the Ancient Greeks. The ancient Greek mathematician and astronomer Oenopides used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gnomon Streckenteilung
A gnomon (; ) is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields. History A painted stick dating from 2300 BC that was excavated at the astronomical site of Taosi is the oldest gnomon known in China. The gnomon was widely used in ancient China from the second century BC onward in order to determine the changes in seasons, orientation, and geographical latitude. The ancient Chinese used shadow measurements for creating calendars that are mentioned in several ancient texts. According to the collection of Zhou Chinese poetic anthologies '' Classic of Poetry'', one of the distant ancestors of King Wen of the Zhou dynasty used to measure gnomon shadow lengths to determine the orientation around the 14th century BC. The ancient Greek philosopher Anaximander (610–546 BC) is credited with introducing this Babylonian instrument to the Ancient Greeks. The ancient Greek mathematician and astronomer Oenopides used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Straightedge And Compass Construction
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass; see compass equivalence theorem. Note however that whilst a non-collapsing compass held against a straightedge might seem to be equivalent to marking it, the neusis construction is still impermissible and this is what unmarked really means: see Markable rulers below.) More formally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, the four concepts—''parallelepiped'' and ''cube'' in three dimensions, ''parallelogram'' and ''square'' in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only ''parallelograms'' and ''parallelepipeds'' exist. Three equivalent definitions of ''parallelepiped'' are *a polyhedron with six faces (hexahedron), each of which is a parallelogram, *a hexahedron with three pairs of parallel faces, and *a prism of which the base is a parallelogram. The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped. "Parallelepiped" is now usually pronounced or ; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space Diagonal
In geometry, a space diagonal (also interior diagonal or body diagonal) of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with ''face diagonals'', which connect vertices on the same face (but not on the same edge) as each other. For example, a pyramid has no space diagonals, while a cube (shown at right) or more generally a parallelepiped has four space diagonals. Axial diagonal An axial diagonal is a space diagonal that passes through the center of a polyhedron. For example, in a cube with edge length ''a'', all four space diagonals are axial diagonals, of common length a\sqrt . More generally, a cuboid with edge lengths ''a'', ''b'', and ''c'' has all four space diagonals axial, with common length \sqrt. A regular octahedron has 3 axial diagonals, of length a\sqrt , with edge length ''a''. A regular icosahedron has 6 axial diagonals of length a\sqrt , where \varphi is the golden ratio (1+\sqrt 5)/2.. Space diagonal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The definition of length (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. In ancient times, volume is measured using similar-shaped natural containers and later on, standardized containers. Some simple three-dimensional shapes can have its volume easily calculated using arithmetic formulas. Volumes of more complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. Zero-, one- and two-dimensional objects have no volume; in fourth and higher dimensions, an analogous concept to the normal vo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
