![Istvan Lenart and spheres](https://upload.wikimedia.org/wikipedia/commons/9/9a/Istvan_Lenart_and_spheres.jpg)
A Lénárt sphere is a educational
manipulative and
writing surface for exploring
spherical geometry, invented by
Hungarian István Lénárt as a modern replacement for a spherical
blackboard.
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- colline ...
s,
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 to ...
s,
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 specia ...
s, and other objects on a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
, 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 wit ...
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
![Lenart sphere equilateral triangle](https://upload.wikimedia.org/wikipedia/commons/9/93/Lenart_sphere_equilateral_triangle.jpg)
The Lénárt sphere and accessories are produced by the company Lénárt Educational Research and Technology.
The basic set includes:
* An eight-inch transparent plastic
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
* 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 tou ...
-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 source ...
s or cut out shapes with
scissors
* A spherical ruler with two scaled edges for drawing
great-circle arcs and measuring
spherical angles and
great-circle distances
* A spherical compass and
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
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 marker Wet-wipe markers or wet-erase markers are a type of writing implement, which are used primarily on overhead transparencies, tablets at restaurants, and office calendars. Other uses include writing on mirrors, chalkboards, plastics, ceramics, glass w ...
s 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 marker Wet-wipe markers or wet-erase markers are a type of writing implement, which are used primarily on overhead transparencies, tablets at restaurants, and office calendars. Other uses include writing on mirrors, chalkboards, plastics, ceramics, glass w ...
s, and Lénárt's book ''Non-Euclidean Adventures on the Lenart Sphere'', which describes more activities for students.
Related tools
Spherical Easel
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
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 celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
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, and
geodesy 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 i ...
, 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 re ...
. 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 g ...
and
geographic information systems (GIS).
See also
*
Blackboard
*
Whiteboard
A whiteboard (also known by the terms marker board, dry-erase board, dry-wipe board, and pen-board) is a glossy, usually white surface for making non-permanent markings. Whiteboards are analogous to blackboards, but with a smoother surface all ...
*
Spherical geometry
*
Celestial navigation
*
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 re ...
*
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 g ...
References
External links
Official website lenartsphere.com
Spherical Easel interactive spherical geometry software
Profile of István Lénártat
ResearchGate with some of Lénárt's research papers
{{DEFAULTSORT:Lenart sphere
Trigonometry
Non-Euclidean geometry
Spherical geometry
Spherical trigonometry
Projective geometry
Geometry education