In geometry, the circumference of a circle is the distance around it. That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge of a disk, circumference is a special case of perimeter; the perimeter is the length around any closed figure and is the term used for most figures excepting the circle and some circular-like figures such as ellipses. Informally, "circumference" may refer to the edge itself rather than to the length of the edge; the circumference of a circle is the distance around it, but if, as in many elementary treatments, distance is defined in terms of straight lines, this cannot be used as a definition. Under these circumstances, the circumference of a circle may be defined as the limit of the perimeters of inscribed regular polygons as the number of sides increases without bound; the term circumference is used when measuring physical objects, as well as when considering abstract geometric forms.

The circumference of a circle is related to one of the most important mathematical constants. This constant, pi, is represented by the Greek letter π; the first few decimal digits of the numerical value of π are 3.141592653589793... Pi is defined as the ratio of a circle's circumference C to its diameter d: π = C d. Or, equivalently, as the ratio of the circumference to twice the radius; the above formula can be rearranged to solve for the circumference: C = π ⋅ d = 2 π ⋅ r. The use of the mathematical constant π is ubiquitous in mathematics and science. In Measurement of a Circle written circa 250 BCE, Archimedes showed that this ratio was greater than 310/71 but less than 31/7 by calculating the perimeters of an inscribed and a circumscribed regular polygon of 96 sides; this method for approximating π was used for centuries, obtaining more accuracy by using polygons of larger and larger number of sides. The last such calculation was performed in 1630 by Christoph Grienberger who used polygons with 1040 sides.

Circumference is used by some authors to denote the perimeter of an ellipse. There is no general formula for the circumference of an ellipse in terms of the semi-major and semi-minor axes of the ellipse that uses only elementary functions. However, there are approximate formulas in terms of these parameters. One such approximation, due to Euler, for the canonical ellipse, x 2 a 2 + y 2 b 2 = 1, is C e l l i p s e ∼ π 2; some lower and upper bounds on the circumference of the canonical ellipse with a ≥ b are 2 π b ≤ C ≤ 2 π a, π ≤ C ≤ 4, 4 a 2 + b 2 ≤ C ≤ π 2. Here the upper bound 2 π a is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, the lower bound 4 a 2 + b 2 is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and minor axes; the circumference of an ellipse can be expressed in terms of the complete elliptic integral of the second kind. More we have C e l l i p s e = 4 a ∫ 0 π / 2 1 − e 2 sin 2 ⁡ θ d θ, where again a is the length of the semi-major axis and e is the eccentricity 1 − b 2 / a 2.

In graph theory the circumference of a graph refers to the longest cycle contained in that graph. Arc length Area Isoperimetric inequality Numericana - Circumference of an ellipse

Bruce Billings

Bruce Billings is an American former professional baseball pitcher. He played in Major League Baseball for the Colorado Rockies, Oakland Athletics, New York Yankees, he pitched in the Chinese Professional Baseball League for the Uni-President 7-Eleven Lions and Fubon Guardians. Billings is the son of Emily. Billings attended the San Diego School of Creative and Performing Arts and Samuel F. B. Morse High School and San Diego State University who played for the SDSU baseball team as a short reliever in 2004 and a starter in 2005 and 2006. In 2005, he played collegiate summer baseball with the Wareham Gatemen of the Cape Cod Baseball League. While at SDSU, Bruce Billings became the school's all-time strikeout leader. Billings was drafted by the Philadelphia Phillies in the 2006 MLB June Amateur Draft in the 31st round, but decided to stay one more year at San Diego State. Billings was again drafted by the Colorado Rockies in the 30th round in the 2007 MLB June Amateur Draft, 912th overall, he accepted the deal.

Billings started his minor league career with the Tri-City Dust Devils where he went 4-2 in 15 starts. In 2008 with the Asheville Tourists, Billings threw a no-hitter against the Lakewood BlueClaws. Billings started to convert to a reliever in 2010 with Tulsa with more than half of his appearances being in relief. Billings was a mid-season All Star with Tulsa. Billings did not pitch one start in 2011 with Colorado Springs before being promoted. While in AA Tulsa, he set the team record for most consecutive scoreless innings at 38. On May 25, 2011, Billings was recalled to the majors when Jorge de la Rosa being placed on the 15-day disabled list with a tear of the ulnar collateral ligament in his left elbow. On May 27, Billings made his major league debut in relief of Matt Daley. Billings pitched the 8th and the 9th innings in a 10-3 loss to the St. Louis Cardinals, giving up 5 hits and 1 run, while recording no strikeouts. Billings and a player to be named were traded to the Oakland Athletics for Mark Ellis on June 30, 2011.

He pitched five innings in three games for the Athletics. He spent the next two seasons with the Sacramento River Cats in AAA. While in AAA for the Oakland A’s, Billings was voted Most Valuable Pitcher in 2012, was runner up in 2013 before electing free agency in 2014. Billings signed a minor league deal with the New York Yankees in December 2013, he appeared for the AAA Scranton/Wilkes-Barre RailRiders and one for the Yankees, where he allowed four runs in four innings, while recording 7 strikeouts. After the game, he found a muscle hernia in his forearm, he was designated for assignment on July 22, 2014. He was released on August 2. On August 7, 2014, he signed a minor league contract with the Los Angeles Dodgers and reported to the AAA Albuquerque Isotopes, he appeared in five games for the Isotopes, all as a relief pitcher and was 1-1 with a 6.75 ERA. He became a free agent at the end of the 2014 season. On November 21, 2014, the Washington Nationals signed Billings to a minor league deal with an invitation to spring training.

In AAA Syracuse, he went 8-5 while posting a 3.63 ERA. He recorded 90 strikeouts in 121.1 innings with a WHIP of 1.26. He elected free agency on November 6, 2015. In February 2016, he signed with the Uni-President 7-Eleven Lions, a professional baseball team based in Taiwan, he joined the spring training in mid-February with a former MLB players, Felix Pie, Jair Jurrjens. He became the ACE of staff over the course of the season, leading the team in Wins, innings, complete games, Whip. Here is an article where Billings is interviewed about the league dynamics for pitchers ( He re-signed with the team for the 2017 season, again leading the team in WINs, strikeouts and Whip, he became a free agent following the season because the team did not want to guarantee him a contract through the season. On March 4, 2018, Billings signed with the Fubon Guardians of the Chinese Professional Baseball League. In late 2018, Billings announced his retirement in order to pursue a coaching opportunity in the Philadelphia Phillies organization.

Career statistics and player information from MLB, or ESPN, or Baseball-Reference, or Fangraphs, or Baseball-Reference San Diego State Aztecs bio


A pseudoscope is a binocular optical instrument that reverses depth perception. It is used to study human stereoscopic perception. Objects viewed through it appear inside out, for example: a box on a floor would appear as a box shaped hole in the floor, it uses sets of optical prisms, or periscopically arranged mirrors to swap the view of the left eye with that of the right eye. In the 1800s Charles Wheatstone coined the name from the Greek ψευδίς σκοπειν -- "false view"; the device was used to explore his theory of stereo vision. Pseudoscopic is 3D in reverse; that is, in aerial photography, swimming pools appear to look like buildings and buildings appear to look like swimming pools. In red and green plotters like the Kelsh and Multiplex this is achieved by reversing the lenses on the 3D glasses; the images will be reverse order. The right image will be viewed through the left eye, the left image will be through the right eye. Switching the two pictures in a standard stereoscope changes all the elevated parts into depressions, vice versa.

The pseudoscope changes convex into concave, high-relief into low-relief. Before the pseudoscope itself was created intentionally, it existed in binocular instruments as an imperfection; the first binocular microscope was invented by the Capuchin monk Cherubin d'Orleans. Because his instrument consisted of two inverting systems, it produced a pseudoscopic impression of depth by accident, although not recognized by microscopists of the time; the instrument subsequently fell into complete neglect for nearly two centuries. It was revived in 1852 by Charles Wheatstone, who published his ideas in his paper "On Binocular Vision," in the Philosophical Transactions for 1852. Wheatstone's paper stimulated the investigation of binocular vision and many variations of pseudoscopes were created, chief types being the mirror or the prismatic. In 1853 the American scientist John Leonard Riddell devised his binocular microscope, which contained the essentials of Wheatstone's pseudoscope. Further reading & commercial pseudoscope Make your own Pseudoscope for 10 dollars Specifications of portable pseudoscope for long term wearing History of design prismatic pseudoscope