A Lénárt sphere is an educational manipulative and

writing surface A writing material, also called a writing medium, is a surface that can be written on with writing instruments, suitable instruments, or used for symbolic or representational drawings. Building materials on which writings or drawings are produce ...

for exploring

spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry or spherics () is the geometry of the two-dimensional surface of a sphere or the -dimensional surface of higher dimensional spheres. Long studied for its practical applicati ...

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

blackboard A blackboard or a chalkboard is a reusable writing surface on which text or drawings are made with sticks of calcium sulphate or calcium carbonate, better known as chalk. Blackboards were originally made of smooth, thin sheets of black or da ...

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

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

s, conics, and other objects on a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, 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 mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

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

ruler A ruler, sometimes called a rule, scale, line gauge, or metre/meter stick, is an instrument used to make length measurements, whereby a length is read from a series of markings called "rules" along an edge of the device. Usually, the instr ...

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 No ...

support synthetic straightedge and compass construction 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 (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

-shaped support to place under the sphere * Hemispherical transparencies that fit over the sphere for students to draw on with marker pens or cut out shapes with

scissors Scissors are hand-operated shearing tools. A pair of scissors consists of a pair of 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 cutting var ...

* 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 between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path between the two points on the surface of the ...

s * A spherical compass and center locator for drawing small circles * 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 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 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 edge (geometry), sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, ge ...

is fundamental to ancient

astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...

and

astrology Astrology is a range of Divination, divinatory practices, recognized as pseudoscientific since the 18th century, that propose that information about human affairs and terrestrial events may be discerned by studying the apparent positions ...

, celestial navigation, and

geodesy Geodesy or geodetics is the science of measuring and representing the Figure of the Earth, geometry, Gravity of Earth, gravity, and Earth's rotation, spatial orientation of the Earth in Relative change, temporally varying Three-dimensional spac ...

and

cartography Cartography (; from , '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 imagined reality) can ...

, 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 method, scholarly research into the transfer of mathematical know ...

. 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 and

geographic information system A geographic information system (GIS) consists of integrated computer hardware and Geographic information system software, software that store, manage, Spatial analysis, analyze, edit, output, and Cartographic design, visualize Geographic data ...

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