topology

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Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

The torus is not simply connected. While the small loop c can be shrunk to a point without breaking the loop or the torus, loops a and b cannot because they encompass the torus's central hole.

Understand the hairy ball theorem topology

In knot theory, knots are formed by seamlessly merging the ends of a segment to form a closed loop. Knots are then characterized by the number of times and the manner in which the segment crosses itself. After the basic loop, the simplest knot is the trefoil knot, which is the only knot, other than its mirror image, that can be formed with exactly three crossings.

The topological concept of a continuous functionA function f from a topological space X to a topological space Y is continuous at p ∊ X if, for any neighbourhood V of f(p), there exists a neighbourhood U of p such that f(U) ⊆ V.

Mathematical knotsKnots are characterized by the number of times and the manner in which the strand crosses itself. A basic loop, which has no crossings and forms only one distinct “knot,” is the simplest knot in knot theory. The number of distinct knots greatly increases with the number of crossings; only those with seven or fewer crossings are shown here.

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In this aerial photo, structures are damaged and destroyed October 15, 2005 in Balakot, Pakistan. It is estimated that 90% of the city of Balakot was leveled by the earthquake. The death toll in the 7.6 magnitude earthquake that struck northern Pakistan on October 8, 2005 is believed to be 38,000 with at least 1,300 more dead in Indian Kashmir. SEE CONTENT NOTES.

6 of the World’s Deadliest Natural Disasters

Remarkable new details of Pluto's largest moon Charon are revealed in this image from New Horizons' Long Range Reconnaissance Imager (LORRI), taken late on July 13, 2015 from a distance of 289,000 miles

Why Is Pluto No Longer a Planet?

Horses. Equus caballus. Horse stable. A brown horse looks out from his stall through the window.

Horsing Around: 7 of the Weirdest Racehorse Names in History

Shah Jahan. Taj Mahal. Mughal architecture. Emperor Shah Jahan fifth Mughal Emperor (reigned 1628-1658) India, Himachal Pradesh, Basohli or Jammu and Kashmir, Mankot, circa 1690 Drawings; Opaque watercolor, gold, and ink on paper (see notes)

6 Important Mughal Emperors

bird. mourning dove. pigeon and dove. Mourning Dove (Zenaida macroura) family Columbidae.

Nikola Tesla's Weird Obsession with Pigeons

Sailboat against a beautiful landscape

Which Waters Do You Pass Through When You “Sail the Seven Seas”?

Data analysis of the Super Bowl's "winningest" teams, game locations, MVP winners by position, teams with the most game losses. football, sports, infographic

From Sport to Spectacle: The History of the Super Bowl

topology: Media

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Understand the hairy ball theorem topology

Understand the hairy ball theorem topology

Learn about the hairy ball theorem of topology.

Video: © MinutePhysics (A Britannica Publishing Partner)

Images

Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically,...

Encyclopædia Britannica, Inc.

The torus is not simply connected. While the small loop c can be shrunk to a point without breaking the loop or the torus, loops a and b cannot because they encompass the torus's central hole.

The torus is not simply connected. While the small loop c can be shrunk...

Encyclopædia Britannica, Inc.

In knot theory, knots are formed by seamlessly merging the ends of a segment to form a closed loop. Knots are then characterized by the number of times and the manner in which the segment crosses itself. After the basic loop, the simplest knot is the trefoil knot, which is the only knot, other than its mirror image, that can be formed with exactly three crossings.

In knot theory, knots are formed by seamlessly merging the ends of a segment to form...

Encyclopædia Britannica, Inc.

The topological concept of a continuous functionA function f from a topological space X to a topological space Y is continuous at p ∊ X if, for any neighbourhood V of f(p), there exists a neighbourhood U of p such that f(U) ⊆ V.

The topological concept of a continuous functionA...

Encyclopædia Britannica, Inc.

Mathematical knotsKnots are characterized by the number of times and the manner in which the strand crosses itself. A basic loop, which has no crossings and forms only one distinct “knot,” is the simplest knot in knot theory. The number of distinct knots greatly increases with the number of crossings; only those with seven or fewer crossings are shown here.

Mathematical knotsKnots are characterized...

Encyclopædia Britannica, Inc.

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