A Lénárt sphere is a educational manipulative and writing surface for exploring

spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...

, invented by Hungarian István Lénárt as a modern replacement for a spherical

blackboard A blackboard (also known as a chalkboard) is a reusable writing surface on which text or drawings are made with sticks of calcium sulphate or calcium carbonate, known, when used for this purpose, as chalk. Blackboards were originally made of ...

. It can be used for visualizing the geometry of points, great and small circles,

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...

conic In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a sp ...

s, and other objects on a

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

, and comparing spherical geometry to

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

as drawn on a flat piece of paper or blackboard. The included spherical

ruler A ruler, sometimes called a rule, line gauge, or scale, is a device used in geometry and technical drawing, as well as the engineering and construction industries, to measure distances or draw straight lines. Variants Rulers have long ...

and

compass A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with ...

support synthetic

straightedge and compass construction In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

on the sphere.

Products

The Lénárt sphere and accessories are produced by the company Lénárt Educational Research and Technology. The basic set includes: * An eight-inch transparent plastic

* A

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not ...

-shaped support to place under the sphere * Hemispherical transparencies that fit over the sphere for students to draw on with

marker pen A marker pen, fine liner, marking pen, felt-tip pen, felt pen, flow marker, sign pen (in South Korea), vivid (in New Zealand), texta (in Australia), sketch pen (in South Asia) or koki (in South Africa), is a pen which has its own ink sourc ...

s or cut out shapes with

scissors Scissors are hand-operated shearing tools. A pair of scissors consists of a pair of metal blades pivoted so that the sharpened edges slide against each other when the handles (bows) opposite to the pivot are closed. Scissors are used for cutt ...

* A spherical ruler with two scaled edges for drawing great-circle arcs and measuring spherical angles and

great-circle distance The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle. It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a st ...

s * A spherical compass and center locator for drawing

small circle A circle of a sphere is a circle that lies on a sphere. Such a circle can be formed as the intersection of a sphere and a plane, or of two spheres. Circles of a sphere are the spherical geometry analogs of generalised circles in Euclidean space ...

s * A set of wet-wipe markers for writing and drawing on the sphere and transparencies * A hanger for displaying spherical constructions and designs * A 16-page booklet of suggested activities, "Getting Started on the Lenart Sphere" * A four-color

polyconic projection The American polyconic map projection is a map projection used for maps of the United States and regions of the United States beginning early in the 19th century. It belongs to the polyconic projection class, which consists of map projections wh ...

of the earth that one can cut out and transform into a globe The company also sells replacement parts, extra transparency sheets, wet-wipe markers, and Lénárt's book ''Non-Euclidean Adventures on the Lenart Sphere'', which describes more activities for students.

Related tools

Spherical Easel is an

interactive geometry software Interactive geometry software (IGS) or dynamic geometry environments (DGEs) are computer programs which allow one to create and then manipulate geometric constructions, primarily in plane geometry. In most IGS, one starts construction by putting a ...

tool for exploring spherical geometry (see ). Other interactive geometry software is typically limited to the flat plane.

History

Spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...

is fundamental to ancient

astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...

and

astrology Astrology is a range of divinatory practices, recognized as pseudoscientific since the 18th century, that claim to discern information about human affairs and terrestrial events by studying the apparent positions of celestial objects. Di ...

celestial navigation Celestial navigation, also known as astronavigation, is the practice of position fixing using stars and other celestial bodies that enables a navigator to accurately determine their actual current physical position in space (or on the surface ...

, and

geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), Earth rotation, orientation in space, and Earth's gravity, gravity. The field also incorporates studies of how these properti ...

and

cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an ...

, and it used to be a standard part of undergraduate

mathematics education In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge. Although ...

. In recent decades hand computations have been replaced by electronic computers and spherical trigonometry has been pushed out of the typical mathematics curriculum by other topics. The Lénárt sphere was invented by István Lénárt in Hungary in the early 1990s and its use is described in his 2003 book comparing planar and spherical geometry. The Lénárt sphere is widely used throughout Europe in university courses on

non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ...

and

geographic information system A geographic information system (GIS) is a type of database containing geographic data (that is, descriptions of phenomena for which location is relevant), combined with software tools for managing, analyzing, and visualizing those data. In a ...

s (GIS).

References

External links

Official website
lenartsphere.com
Spherical Easel
interactive spherical geometry software
Profile of István Lénárt
at

ResearchGate ResearchGate is a European commercial social networking site for scientists and researchers to share papers, ask and answer questions, and find collaborators. According to a 2014 study by ''Nature'' and a 2016 article in '' Times Higher Education ...

with some of Lénárt's research papers {{DEFAULTSORT:Lenart sphere Trigonometry Non-Euclidean geometry Spherical geometry Spherical trigonometry Projective geometry Geometry education

Products

Related tools

History

See also

References

External links