The shape (Old English: gesceap, created thing) of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material composition.

Mathematician and statistician David George Kendall writes:^[1]

In this paper ‘shape’ is used in the vulgar sense, and means what one would normally expect it to mean. [...] We here define ‘shape’ informally as ‘all the geometrical information that remains when location, scale^[2] and rotational effects are filtered out from an object.’

Simple shapes can be described by basic geometry objects such as a set of two or more points, a line, a curve, a plane, a plane figure (e.g. square or circle), or a solid figure (e.g. cube or sphere). Most shapes occurring in the physical world are complex. Some, such as plant structures and coastlines, may be so arbitrary as to defy traditional mathematical description – in which case they may be analyzed by differential geometry, or as fractals.

Rigid shape definition[link]

In geometry, two subsets of a Euclidean space have the same shape if one can be transformed to the other by a combination of translations, rotations (together also called rigid transformations), and uniform scalings. In other words, the shape of a set of points is all the geometrical information that is invariant to translations, rotations, and size changes. Having the same shape is an equivalence relation, and accordingly a precise mathematical definition of the notion of shape can be given as being an equivalence class of subsets of a Euclidean space having the same shape.

Shapes of physical objects are equal if the subsets of space these objects occupy satisfy the definition above. In particular, the shape does not depend on the size and placement in space of the object. For instance, a "p" and a "d" have the same shape, as they can be perfectly superimposed if the "p" is translated to the right by a given distance, rotated upside down and magnified by a given factor (see Procrustes superimposition for details). However, a mirror image could be called a different shape. For instance, a "b" and a "d" have a different shape, at least when they are constrained to move within a two-dimensional space like the page on which they are written. Even though they have the same size, there's no way to perfectly superimpose them by traslating and rotating them along the page. Similarly, within a three-dimensional space, a right hand and a left hand have a different shape, even if they are the mirror images of each other. Shapes may change if the object is scaled non uniformly. For example, a sphere becomes an ellipsoid when scaled differently in the vertical and horizontal directions. In other words, preserving axes of symmetry (if they exist) is important for preserving shapes. Also, shape is determined by only the outer boundary of an object. For example, a solid ice cube and a second ice cube containing an inner cavity (air bubble) have the same shape.^{[citation needed]}

Objects that can be transformed into each other by rigid transformations and mirroring are congruent. An object is therefore congruent to its mirror image (even if it is not symmetric), but not to a scaled version. Objects that have the same shape or one has the same shape as the other's mirror image (or both if they are themselves symmetric) are called geometrically similar. Thus congruent objects are always geometrically similar, but geometrical similarity additionally allows uniform scaling.

Non-rigid shape definition[link]

A more flexible definition of shape takes into consideration the fact that realistic shapes are often deformable, e.g. a person in different postures, a tree bending in the wind or a hand with different finger positions. By allowing also isometric (or near-isometric) deformations like bending, the intrinsic geometry of the object will stay the same, while sub-parts might be located at very different positions in space. This definition uses the fact that, geodesics (curves measured along the surface of the object) stay the same, independent of the isometric embedding. This means that the distance from a finger to a toe of a person measured along the body is always the same, independent of posture. By only considering geodesic distances or other isometric properties as done in spectral shape analysis, it is possible to retrieve all cats in a database of animals independent of the pose.

Colloquial shape definition[link]

Shape can also be more loosely defined as "the appearance of something, especially its outline". This definition is consistent with the above, in that the shape of a set does not depend on its position, size or orientation. However, it does not always imply an exact mathematical transformation. For example it is common to talk of star-shaped objects even though the number of points of the star is not defined.

Philosophical skepticism of definitions[link]

In Plato's Meno, Socrates questions Meno as to the most accurate definition of a figure/shape. By showing the possibility of there being more than one definition, Socrates shows that a definition cannot wholly and accurately describe something and that there is no one absolute definition for anything including shape. ^[3]

Shape analysis[link]

The modern definition of shape^{[clarification needed]} has arisen in the field of statistical shape analysis. In particular Procrustes analysis, which is a technique for analyzing the statistical distributions of shapes. These techniques have been used to examine the alignments of random points. Other methods are designed to work with non-rigid (bendable) objects, e.g. for posture independent shape retrieval (see for example Spectral shape analysis).

References[link]

External links[link]

Look up shape in Wiktionary, the free dictionary.

• Answers for many questions related to shapes and sizes of common objects
• American artist Allan McCollum's project to create a unique "shape" for every individual on the planet
• Shape Stencils from Stencil Ease

Patrick Willis

Willis at 49ers training camp in August 2010
No. 52     San Francisco 49ers
Inside Linebacker
Personal information
Date of birth: (1985-01-25) January 25, 1985 (age 27)
Place of birth: Bruceton, Tennessee
Height: 6 ft 1 in (1.85 m) Weight: 240 lb (109 kg)
Career information
College: Mississippi
NFL Draft: 2007 / Round: 1 / Pick: 11
Debuted in 2007 for the San Francisco 49ers
Career history
San Francisco 49ers (2007–present)
Career highlights and awards
5× Pro Bowl (2007, 2008, 2009, 2010, 2011) 4× AP First-team All-Pro (2007, 2009, 2010, 2011) AP Second-team All-Pro (2008) AP NFL Defensive Rookie of the Year (2007) 3× NFL Alumni Linebacker of the Year (2007, 2009, 2010) 2× Butkus Award (2006, 2009) 2× First-team All-American (2005, 2006) Other College Awards and Honors
Career NFL statistics as of Week 17, 2011
Tackles 692 Quarterback sacks 17.0 Interceptions 5 Forced Fumbles 12 Stats at NFL.com

Patrick L. Willis (born January 25, 1985) is an American football player who plays inside linebacker for the San Francisco 49ers of the National Football League (NFL). He was selected by the San Francisco 49ers 11th overall in the 2007 NFL Draft. He played college football at the University of Mississippi.

During Willis's senior season at Ole Miss, he received the Butkus Award and the Jack Lambert Award as the nation’s top linebacker. A year later as a member of the 49ers, Willis led the NFL in tackles, earned first-team All-Pro and Pro Bowl honors while being named the 2007 AP NFL Defensive Rookie of the Year. Willis has earned Pro Bowl and All-Pro honors in all five years he has played in the NFL. He is the only player to receive the Butkus Award for best linebacker in more than one category. He won the college Butkus Award in 2006 while at Ole Miss and in 2009, he won the professional Butkus Award while with the San Francisco 49ers.^[1]

Patrick Willis is regarded by many as the best inside linebacker in professional football.^[2][3][4]

Early life[link]

Born in Bruceton, Tennessee, Willis attended Hollow Rock-Bruceton Central High School,^[5] where he was a two-time All-State selection, Regional Most Valuable Player, and West Tennessee Player of the Year. He earned 4 letters in football and basketball, and 3 in baseball. He was also the first person in Tennessee state history to be nominated for both the Mr. Football Award for a Lineman (as a linebacker) and the Mr. Football Award for a Back (as a tailback) in the same season.

Regarded as a three-star recruit by Rivals.com, Willis was listed as the No. 60 linebacker prospect in the class of 2003.^[6] He chose Ole Miss over Memphis.

College career[link]

Willis attended the University of Mississippi ("Ole Miss"), and played for the Ole Miss Rebels football team. As a freshman at Ole Miss, Willis played in all 13 games, making 20 stops. He received the Scholar-Athlete Award from the Ole Miss Chapter of the National Football Foundation and College Football Hall of Fame.

As a sophomore in 2004, he appeared in 10 of 11 games and earned honorable mention All-Southeastern Conference (SEC) honors from The Associated Press. He recorded 70 tackles (54 solo), and led the team with 11.0 TFLs (tackles-for-loss) and five sacks.

In Willis's 2005 junior campaign, he led the SEC in total tackles at 12.80 per game, which placed him sixth nationally. He finished season with 128 total tackles, 9.5 TFLs, three sacks, an interception, two forced fumbles and a fumble recovery. Willis was named SEC Defensive Player of the Year by CollegeFootballNews.com^[8] and Scout.com. He was also named as a first-team All-American by CollegeFootballNews.com and the All-American Football Foundation.^[9] Willis earned first-team All-SEC honors from Associated Press, SEC Coaches, CollegeFootballNews.com and Rivals.com.

During Willis's last collegiate year in 2006, he led the SEC in tackles once again with 11.4 per game and collected 137 tackles, 11.5 TFLs, seven passes deflected, three sacks, two forced fumbles and one fumble recovery. He was awarded SEC Defensive Player of the Year, first-team All-SEC, and consensus first-team All-American. He was the winner of the Jack Lambert Award and the prestigious Dick Butkus Award, given to the most outstanding linebacker in college football. He also won the Conerly Trophy, voted upon by the media in Mississippi and awarded to the best college football player in Mississippi. In addition, he was a finalist for the Bronko Nagurski Award and semifinalist for both the Rotary Lombardi Award and the Lott Trophy.^[10]

Awards and Honors[link]

• 2× First-team All-SEC (2005, 2006)
• 2× All-American (2005, 2006)
• SEC Defensive Player of the Year (2006)
• Butkus Award (2006)
• Jack Lambert Award (2006)
• Conerly Trophy (2006)

Professional career[link]

2007 NFL Draft[link]

Originally regarded as a late-first to early-second round draft pick,^[11] Willis improved his draft stock with an impressive performance at the NFL Combine; his 4.51 40-yard dash^[12][13][14] was one of the fastest among linebackers. He also posted a 39-inch vertical jump and recorded 22 repetitions of the 225-lb. bench press. Willis clocked in at 4.37 seconds for his 40-yard dash during his University of Mississippi pro-day workout.^[15] Analysts of The Sporting News compared him to Jeremiah Trotter.^[16] Willis also earned South Team Defensive MVP honors at the 2007 Senior Bowl.

Willis was selected by the San Francisco 49ers with the 11th overall pick in the 2007 NFL Draft, re-uniting him with the staff that coached his South Squad at the 2007 Senior Bowl. He signed a five-year, $16.7 million contract with the 49ers.^[17]

San Francisco 49ers (2007–present)[link]

2007 NFL season

Willis was coached directly by Hall of Fame middle-linebacker/coach Mike Singletary (who later became 49ers head coach), and eventually started at inside linebacker, specifically the "Mike" position, in the 49ers' 3-4 defense. In his first NFL game, Willis recorded 11 tackles, including 9 solo, and a forced fumble as the 49ers defeated the Arizona Cardinals in dramatic fashion by a score of 20-17. For his effort, he won the Diet Pepsi Rookie of the Week Award. One of Willis's strongest performances came against the Minnesota Vikings, as he led the 49ers defense in containing Offensive Rookie of the Year, Adrian Peterson, who managed just 3 yards on 14 carries. Willis's most impressive statistical performance came in week 16 at home against the Tampa Bay Buccaneers. In this game he recorded 20 total tackles, 2 sacks, 1 forced fumble and 1 pass deflected in a 21-19 victory, earning Rookie of the Week honors for the fourth time. In addition to receiving the award in weeks 1 and 16, Willis took home the award in weeks 2 and 12. He was also awarded the GMC Defensive Player of the Week twice.

In a game against the Arizona Cardinals on November 25, Willis ran down wide receiver Sean Morey after a 62-yard, catch-and-run for the game-saving tackle in overtime. The Niners eventually won 37-31.^[18]

Willis finished the 2007 campaign with an NFL-leading 174 total tackles, along with 4 sacks, 2 forced fumbles, and 5 passes deflected. According to the 49ers, however, he is unofficially credited with over 200 tackles during that season. As a result, Willis was one of two 49ers selected to the Pro Bowl along with punter Andy Lee. Other honors earned at the conclusion of his first professional season included being named AP NFL Defensive Rookie of the Year, first-team All-Pro, and NFL Alumni Linebacker Of The Year. He was the only rookie in the NFL to make the AP All-Pro squad and the first defensive rookie from the 49ers to make the Pro Bowl since Ronnie Lott in 1981. His season was capped by being named one of ESPN The Magazine's "NEXT" athletes of 2008, alongside Yankee Joba Chamberlain and Trail Blazer Brandon Roy.

2008 NFL season

Willis in October 2008

In his second NFL season Willis scored his first career NFL touchdown against Seattle, returning an interception for 86 yards. On December 16, during another dominating season with the 49ers, Willis was named the starting inside linebacker for the NFC Pro Bowl team. In just two seasons, he has established himself as one of the top linebackers in the NFL.^[19]

At the end of the year, Willis finished second in tackles and was voted to his second consecutive Pro Bowl, this time as a starter. He was also voted second-team All-Pro, barely missing the first team as Jon Beason of Carolina Panthers collected one more vote than Willis.

2009 NFL season

Patrick Willis started the season strong in the opener against the Arizona Cardinals. He recorded 14 tackles and an interception in a 20-16 win in Arizona. He also had a huge game in week 4 against St. Louis finishing with 8 tackles, 2.5 sacks and an interception return for a 23 yard touchdown.

The biggest improvement in his game from the previous year was the increased sideline-to-sideline range. Willis made 54 tackles on plays labeled as “Wide left” or “Wide right” which accounted for 35.5% of his total tackles. In that area, he led all inside linebackers by a large margin. He was also the league leader in impact plays with a total of 27.^[20]

At the end of the year, Willis accumulated an NFL-leading 152 tackles along with 4 sacks, 3 forced fumbles, 8 pass deflections and 3 interceptions. He has now led the league in tackles in two of his first three seasons. Willis also earned first-team All-Pro receiving 49/50 votes from the AP and was voted to his third consecutive Pro Bowl. However, shortly before the Pro Bowl, which was set in South Florida, Willis decided to play it safe and opted not to play due to a leg injury he sustained. He stayed in South Florida for the game and stood on the sideline in support of the NFC team and 49er teammates Frank Gore, Vernon Davis, Justin Smith and Andy Lee.^[21]

Patrick Willis was named the 2009 NFL Alumni Linebacker of the Year, an award voted on by past NFL linebackers.^[22] Willis also became the second NFL player to win the Butkus Award since it was expanded in 2008 to include high school and professional winners in part to help end anabolic steroid abuse among young athletes. He became the first linebacker to receive the prestigious award in more than one category, both in college and in the pros.

2010 NFL season

On May 4, Willis signed a five-year contract extension worth $50 million, with $29 million in guaranteed money making him the highest paid inside linebacker in the NFL.^[23]

In 2010, Willis strengthened a new dimension of his game, totaling a career-high 6.0 sacks. He played through the season with several injuries, playing a game against the San Diego Chargers with a cast on his broken hand. Willis was ruled out for Week 17 after undergoing a second surgery on his broken right hand. It was the first time that Willis had ever missed a game in his career. Willis has also made a habit of making it to the Pro Bowl. He’s gone all four seasons of his NFL career and even with his tackle numbers down (largely due to offenses scheming against him), Willis still finished with the second-most fan votes for inside linebackers in the entire NFL.^[24] He now joins Hall of Fame safety Ronnie Lott as only 49ers to be selected to the Pro Bowl in each of their first four seasons.

Willis also earned first-team All-Pro from the Associated Press alongside Jerod Mayo of the New England Patriots.^[25] For the third time in his career, and the second year in a row, he was named Linebacker of the Year by the NFL Alumni Association.^[26]

2011 NFL season

Before the start of the 2011 season, Patrick Willis ran away from the field in the voting for the NFL's best linebacker on ESPN.com. "Nobody in the NFL plays their position better than Patrick Willis, and that is saying a lot," said Matt Williamson of Scouts Inc. "He is as good a linebacker as Peyton Manning is a quarterback, as Andre Johnson is a receiver, as Adrian Peterson is a running back. He has no weaknesses."^[3] Even 12-time Pro Bowler Ray Lewis, the dominant linebacker of his era, pointed to Willis as a worthy successor to his undisputed reign. ESPN's Dana Jacobson asked Lewis which young linebacker reminded him of himself. "I just love the way he plays the game," Lewis said. "He plays the game with a fire. He reminds me of myself—a lot, a lot, a lot."^[27] Willis also came in fourth in an ESPN expert vote for the NFL's 10 best defensive players, finishing only behind 2010 AP NFL Defensive Player of the Year Troy Polamalu, Demarcus Ware, and Darrelle Revis.

In 2011, Willis led a top ranked 49ers defense which set a NFL single season record of not allowing a rushing touchdown until week 16. The previous record was held by the 1920 Decatur Staleys, who did not allow a rushing touchdown in a 13 game season.^[28] The 49ers' stingy defense ranked fourth overall and played a huge role in the 49ers finishing with a 13-3 record. This also marks the first time since 2002 the 49ers are division champs and back in the playoffs.

During a game against the St. Louis Rams in week 13, Willis landed awkwardly after missing a tackle and injured his right hamstring.^[29] Despite missing nearly four games due to the hamstring injury, Willis finished the season with 97 tackles along with 2 sacks, 4 forced fumbles, 12 pass deflections and an interception. He also improved his game significantly in coverage recording a career high 12 pass deflections.^[30] Willis was named to his fifth consecutive Pro Bowl and becomes the first player in franchise history to make five Pro Bowl appearances to start a career.^[31] In addition, he earned first-team All-Pro from the Associated Press for the third consecutive year.^[32]

NFL career statistics[link]

Personal[link]

Willis was separated from his biological father at the age of 17, and along with his three siblings moved in with his parental guardians, Willis's high school basketball coach, Chris Finley, and his wife, Julie.^[33]

He holds a degree in criminal justice and envisions a future in law enforcement.^[33]

His uncle, Arthur Willis, was a professional super middleweight boxer who later fought world champion James Toney to a split decision.^[34]

Willis's brother, Detris, drowned in 2006 while swimming with his friends.^[35]

References[link]

External links[link]

Wikimedia Commons has media related to: Patrick Willis

• Official Website
• Patrick Willis on Twitter
• Ole Miss Rebels bio
• San Francisco 49ers bio
• Scout.com profile
• Interview with Michael Tillery of "The Starting Five"
• [2] Article with SignonSanDiego.com