History of geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics. Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today, his book, The Elements is considered the most influential textbook of all time, was known to all educated people in the West until the middle of the 20th century. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are recognizable as the descendants of early geometry; the earliest recorded beginnings of geometry can be traced to early peoples, who discovered obtuse triangles in the ancient Indus Valley, ancient Babylonia from around 3000 BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.

Among these were some sophisticated principles, a modern mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras and the Indian Sulba Sutras around 800 BC contained the first statements of the theorem. Problem 30 of the Ahmes papyrus uses these methods to calculate the area of a circle, according to a rule that the area is equal to the square of 8/9 of the circle's diameter; this assumes that π is 4×2, with an error of over 0.63 percent. This value was less accurate than the calculations of the Babylonians, but was not otherwise surpassed until Archimedes' approximation of 211875/67441 = 3.14163, which had an error of just over 1 in 10,000. Ahmes knew of the modern 22/7 as an approximation for π, used it to split a hekat, hekat x 22/x x 7/22 = hekat. Problem 48 involved using a square with side 9 units; this square was cut into a 3x3 grid.

The diagonal of the corner squares were used to make an irregular octagon with an area of 63 units. This gave a second value for π of 3.111... The two problems together indicate a range of values for π between 3.11 and 3.16. Problem 14 in the Moscow Mathematical Papyrus gives the only ancient example finding the volume of a frustum of a pyramid, describing the correct formula: V = 1 3 h where a and b are the base and top side lengths of the truncated pyramid and h is the height; the Babylonians may have known the general rules for measuring volumes. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π is estimated as 3; the volume of a cylinder was taken as the product of the base and the height, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases. The Pythagorean theorem was known to the Babylonians. There was a recent discovery in which a tablet used π as 3 and 1/8.

The Babylonians are known for the Babylonian mile, a measure of distance equal to about seven miles today. This measurement for distances was converted to a time-mile used for measuring the travel of the Sun, representing time. There have been recent discoveries showing that ancient Babylonians may have discovered astronomical geometry nearly 1400 years before Europeans did; the Indian Vedic period had a tradition of geometry expressed in the construction of elaborate altars. Early Indian texts on this topic include the Śulba Sūtras. According to, the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had been known to the Old Babylonians." The diagonal rope of an oblong produces both which the flank and the horizontal <ropes> produce separately." They contain lists of Pythagorean triples. They contain statements about squaring the circle and "circling the square."The Baudhayana Sulba Sutra, the best-known and oldest of the Sulba Sutras contains examples of simple Pythagorean triples, such as:, {\displaystyle

List of Harper's Bazaar cover models

The List of Harper's Bazaar cover models gives the models for the covers of Harper's Bazaar magazine. The magazine has different editions in different countries, the list is broken down by country. List of Harper's Bazaar Arabia cover models List of Harper's Bazaar Argentina cover models List of Harper's Bazaar Australia cover models List of Harper's Bazaar Brazil cover models List of Harper's Bazaar Chile cover models List of Harper's Bazaar Germany cover models List of Harper's Bazaar India cover models List of Harper's Bazaar Indonesia cover models List of Harper's Bazaar Japan cover models List of Harper's Bazaar Netherlands cover models List of Harper's Bazaar Poland cover models List of Harper's Bazaar Russia cover models List of Harper's Bazaar Serbia cover models List of Harper's Bazaar Spain cover models List of Harper's Bazaar UK cover models List of Harper's Bazaar Ukraine cover models List of Harper's Bazaar US cover models

Dungeon Floors

Dungeon Floors is a fantasy role-playing game supplement published by Heritage USA in 1981. Dungeon Floors is a set of cards which can be cut up to provide wood and stone floors, etc. for gaming miniatures. Dungeon Floors was composed of a set of nine cardboard sheets which could be used to create tiles that could be arranged to represent different layouts for role-playing spaces, such as building interiors, underground passages. There was an instruction folder describing how to use the tiles and how to integrate the marked spaces on them with common RPG rulesets; the spaces were sized to take 25mm miniatures, such as those made by Heritage and most other companies at the time. There were three editions of Dungeon Floors; the first was a shrink-wrapped group of thin card pieces printed with black ink only. The second edition was printed in two-color format; the third edition was boxed, printed in full-color with detailed realistic artwork. The first two editions were part of Heritage's Dungeon Builders line of products, which featured molded 3-D walls with stone and other textures.

The third and best-selling edition of Dungeon Floors was a standalone product, the Dungeon Builders line having been discontinued by that time. Steve Jackson reviewed Dungeon Floors in The Space Gamer No. 49. Jackson commented that "if you like such things, Dungeon Floors ought to please you greatly."