Alphabet of Lines: Geometric Construction Essay

806 Words Jan 15th, 2013 4 Pages
In antiquity, geometric constructions of figures and lengths were restricted to the use of only a straightedge and compass (or in Plato's case, a compass only; a technique now called a Mascheroni construction). Although the term "ruler" is sometimes used instead of "straightedge," the Greek prescription prohibited markings that could be used to make measurements. Furthermore, the "compass" could not even be used to mark off distances by setting it and then "walking" it along, so the compass had to be considered to automatically collapse when not in the process of drawing a circle.

Because of the prominent place Greek geometric constructions held in Euclid's Elements, these constructions are sometimes also known as Euclidean
Call the upper endpoint of this perpendicular diameter . For the pentagon, find the midpoint of and call it . Draw , and bisect , calling the intersection point with . Draw parallel to , and the first two points of the pentagon are and . The construction for the heptadecagon is more complicated, but can be accomplished in 17 relatively simple steps. The construction problem has now been automated (Bishop 1978).

Simple algebraic operations such as , , (for a rational number), , , and can be performed using geometric constructions (Bold 1982, Courant and Robbins 1996). Other more complicated constructions, such as the solution of Apollonius' problem and the construction of inverse points can also accomplished.

One of the simplest geometric constructions is the construction of a bisector of a line segment, illustrated above.

The Greeks were very adept at constructing polygons, but it took the genius of Gauss to mathematically determine which constructions were possible and which were not. As a result, Gauss determined that a series of polygons (the smallest of which has 17 sides; the heptadecagon) had constructions unknown to the Greeks. Gauss showed that the constructible polygons (several of which are illustrated above) were closely related to numbers called the Fermat primes.

Wernick (1982) gave a list of 139 sets of three located points from which a triangle was to be

Related Documents

• Visual Analysis Paper FINAL

Karen Ibrahim ARH131 – Q Joel Hollander 3/2/15 Visual Analysis Paper The Dark Age of Greece refers to a period of time where all knowledge of Mycenae construction, painting, sculpture, reading and writing disintegrated due to Dorian invasion. One form of art that continued through the period of the Mycenaean downfall was the art of pottery. Pottery, more specifically vase painting, was the beginning of an illustrative development in Greek art and was considered a durable form of art because it…

Words: 2032 - Pages: 9
• The Number Line Essay

4460 01 THE NUMBER LINE BY George L Williams III Contents * THE NUMBER LINE * Extended real number line * Drawing the number line * Line segmentation * Comparing numbers * Arithmetic Operations * Arithmetic Operations (cont.) * Algebraic properties * Cartesian Plane/Cartesian Coordinate System * An Overview * My words * Applications of the number line * Resources * THE NUMBER LINE Mathematics is one of…

Words: 3861 - Pages: 16
• Elements of Montessori Education Essay example

own individual furniture (own mats & chowki) 4. Furniture should be light which promotes freedom of movement. Chairs should be at different heights according to ages. The feet of the children should touch the floor. Table can be of different geometric shapes. 5. Mats should be of different sizes & colors to accommodate the children’s work. They should be in one light color, made of durable material that can be cleaned easily 6. Have different areas for different activities e.g. reading…

Words: 3519 - Pages: 15
• Essay on Haru

The particular advantage or usage of modern robots is in the workplace on product lines. They are used exclusively to make production more efficient and productive for the company, cheaper for the customers and safer for the employees. Robots are widely used in such industries as automobile manufacture to perform simple repetitive tasks…

Words: 5651 - Pages: 23
• Pdf, Docx Essay

Bayes theorem. 4. Independence of two events. Mutual independence of n events. Sampling with and without replacement. 5. Random variables. Univariate distributions - discrete, continuous, mixed. Standard distributions - hypergeometric, binomial, geometric, Poisson, uniform, normal, exponential. Probability mass function, density function, distribution function. Probabilities of events in terms of random variables. 6. Transformations of a single random variable. Mean, variance, median, quantiles. 7…

Words: 29782 - Pages: 120
• Machine Learning Essay

218 7 On-Line Learning 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . 7.2 Prediction with expert advice . . . . . . . . . . . 7.2.1 Mistake bounds and Halving algorithm . 7.2.2 Weighted majority algorithm . . . . . . . 7.2.3 Randomized weighted majority algorithm 7.2.4 Exponential weighted average algorithm . 7.3 Linear classiﬁcation . . . . . . . . . . . . . . . . . 7.3.1 Perceptron algorithm . . . . . . . . . . . . 7.3.2 Winnow algorithm . . . . . . . . . . . . . 7.4 On-line to batch…

Words: 137814 - Pages: 552
• Computer Vision Essay

403 12. Projection and 3D Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Projections Affine and Perspective Transformations POSIT: 3D Pose Estimation Stereo Imaging Structure from Motion Fitting Lines in Two and Three Dimensions Exercises 405 407 412 415 453 454 458 13. Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 What Is Machine Learning Common Routines in the ML Library…

Words: 150680 - Pages: 603

these areas. Dr. Rosen has also worked with several publishing companies on their homework delivery platforms. At Bell Laboratories and AT&T Laboratories, Dr. Rosen worked on a wide range of projects, including operations research studies, product line planning for computers and data communications equipment, and technology assessment. He helped plan AT&T’s products and services in the area of multimedia, including video communications, speech recognition, speech synthesis, and image networking.…

Words: 460915 - Pages: 1844