A shape or figure is a
graphical representation
Graphic communication as the name suggests is communication using graphic elements. These elements include symbols such as glyphs and icons, images such as drawings and photographs, and can include the passive contributions of substrate, colour ...
of an object or its external boundary, outline, or external
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
, as opposed to other properties such as
color
Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are assoc ...
,
texture
Texture may refer to:
Science and technology
* Surface texture, the texture means smoothness, roughness, or bumpiness of the surface of an object
* Texture (roads), road surface characteristics with waves shorter than road roughness
* Texture ...
, or
material
Material is a substance or mixture of substances that constitutes an object. Materials can be pure or impure, living or non-living matter. Materials can be classified on the basis of their physical and chemical properties, or on their geolo ...
type.
A plane shape or plane figure is constrained to lie on a ''
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
'', in contrast to ''
solid
Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structural ...
'' 3D shapes.
A two-dimensional shape or two-dimensional figure (also: 2D shape or 2D figure) may lie on a more general curved ''
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
'' (a non-Euclidean two-dimensional space).
Classification of simple shapes
![Polygon types](https://upload.wikimedia.org/wikipedia/commons/e/eb/Polygon_types.svg)
Some simple shapes can be put into broad categories. For instance,
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 are classified according to their number of edges as
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,
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
s,
pentagons, etc. Each of these is divided into smaller categories; triangles can be
equilateral
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
,
isosceles
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
,
obtuse,
acute
Acute may refer to:
Science and technology
* Acute angle
** Acute triangle
** Acute, a leaf shape in the glossary of leaf morphology
* Acute (medicine), a disease that it is of short duration and of recent onset.
** Acute toxicity, the adverse eff ...
,
scalene, etc. while quadrilaterals can be
rectangles,
rhombi
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
,
trapezoids
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium ().
A trapezoid is necessarily a convex quadrilateral in Eucli ...
,
squares
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
, etc.
Other common shapes are
points,
lines
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
,
planes, and
conic sections such as
ellipses,
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
s, and
parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
s.
Among the most common 3-dimensional shapes are
polyhedra
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on ...
, which are shapes with flat faces;
ellipsoids, which are egg-shaped or sphere-shaped objects;
cylinders; and
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
s.
If an object falls into one of these categories exactly or even approximately, we can use it to describe the shape of the object. Thus, we say that the shape of a
manhole cover
A manhole cover or maintenance hole cover is a removable plate forming the lid over the opening of a manhole, an opening large enough for a person to pass through that is used as an access point for an underground vault or pipe. It is designed t ...
is a
disk, because it is approximately the same geometric object as an actual geometric disk.
In geometry
![Basic shapes](https://upload.wikimedia.org/wikipedia/commons/3/38/Basic_shapes.svg)
A geometric shape consists of the
geometric information which remains when
location
In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...
,
scale,
orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
and
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
are removed from the description of a
geometric object
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
. That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in a mirror is the same shape as the original, and not a distinct shape.
Many two-dimensional geometric shapes can be defined by a set of
points or
vertices and
lines
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
connecting the points in a closed chain, as well as the resulting interior points. Such shapes are called
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 and include
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,
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
s, and
pentagons. Other shapes may be bounded by
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s such as the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
or the
ellipse.
Many three-dimensional geometric shapes can be defined by a set of vertices, lines connecting the vertices, and two-dimensional
faces
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
enclosed by those lines, as well as the resulting interior points. Such shapes are called
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on ...
s and include
cubes as well as
pyramids
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
such as
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
s. Other three-dimensional shapes may be bounded by curved surfaces, such as the
ellipsoid and the
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 shape is said to be
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
if all of the points on a line segment between any two of its points are also part of the shape.
Properties
There are several ways to compare the shapes of two objects:
*
Congruence: Two objects are congruent if one can be transformed into the other by a sequence of rotations, translations, and/or reflections.
*
Similarity: Two objects are similar if one can be transformed into the other by a uniform scaling, together with a sequence of rotations, translations, and/or reflections.
*
Isotopy: Two objects are isotopic if one can be transformed into the other by a sequence of deformations that do not tear the object or put holes in it.
Sometimes, two similar or congruent objects may be regarded as having a different shape if a reflection is required to transform one into the other. For instance, the letters "b" and "d" are a reflection of each other, and hence they are congruent and similar, but in some contexts they are not regarded as having the same shape. Sometimes, only the outline or external boundary of the object is considered to determine its shape. For instance, a hollow sphere may be considered to have the same shape as a solid sphere.
Procrustes analysis is used in many sciences to determine whether or not two objects have the same shape, or to measure the difference between two shapes. In advanced mathematics,
quasi-isometry In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. ...
can be used as a criterion to state that two shapes are approximately the same.
Simple shapes can often be classified into basic
geometric objects such as a
point, a
line, a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
, a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
, a
plane figure
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie o ...
(e.g.
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
or
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
), or a solid figure (e.g.
cube or
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 ...
). However, most shapes occurring in the physical world are complex. Some, such as plant structures and coastlines, may be so complicated as to defy traditional mathematical description – in which case they may be analyzed by
differential geometry, or as
fractals.
Equivalence of shapes
In geometry, two subsets of a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
have the same shape if one can be transformed to the other by a combination of
translations
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
,
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.
Mathematician and statistician
David George Kendall
David George Kendall FRS (15 January 1918 – 23 October 2007) was an English statistician and mathematician, known for his work on probability, statistical shape analysis, ley lines and queueing theory. He spent most of his academic li ...
writes:
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 and rotational effects are filtered out from an object.’
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 "
d" and a "
p" have the same shape, as they can be perfectly superimposed if the "
d" is translated to the right by a given distance, rotated upside down and magnified by a given factor (see
Procrustes superimposition
In statistics, Procrustes analysis is a form of statistical shape analysis used to analyse the distribution of a set of shapes. The name ''Procrustes'' ( el, Προκρούστης) refers to a bandit from Greek mythology who made his victims fi ...
for details). However, a
mirror image could be called a different shape. For instance, a "
b" and a "
p" 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 translating 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
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 ...
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.
Congruence and similarity
Objects that can be transformed into each other by rigid transformations and mirroring (but not scaling) are
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
. An object is therefore congruent to its
mirror image (even if it is not symmetric), but not to a scaled version. Two congruent objects always have either the same shape or mirror image shapes, and have the same size.
Objects that have the same shape or mirror image shapes are called
geometrically similar, whether or not they have the same size. Thus, objects that can be transformed into each other by rigid transformations, mirroring, and uniform scaling are similar. Similarity is preserved when one of the objects is uniformly scaled, while congruence is not. Thus, congruent objects are always geometrically similar, but similar objects may not be congruent, as they may have different size.
Homeomorphism
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.
One way of modeling non-rigid movements is by
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
s. Roughly speaking, a homeomorphism is a continuous stretching and bending of an object into a new shape. Thus, a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
and a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
are homeomorphic to each other, but 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 a
donut
A doughnut or donut () is a type of food made from leavened fried dough. It is popular in many countries and is prepared in various forms as a sweet snack that can be homemade or purchased in bakeries, supermarkets, food stalls, and fra ...
are not. An often-repeated
mathematical joke
A mathematical joke is a form of humor which relies on aspects of mathematics or a stereotype of mathematicians. The humor may come from a pun, or from a double meaning of a mathematical term, or from a lay person's misunderstanding of a mathem ...
is that topologists cannot tell their coffee cup from their donut, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle.
A described shape has external lines that you can see and make up the shape. If you were putting you coordinates on and coordinate graph you could draw lines to show where you can see a shape, however not every time you put coordinates in a graph as such you can make a shape. This shape has a outline and boundary so you can see it and is not just regular dots on a regular paper.
Shape analysis
The above-mentioned mathematical definitions of rigid and non-rigid shape have arisen in the field of
statistical shape analysis. In particular,
Procrustes analysis is a technique used for comparing shapes of similar objects (e.g. bones of different animals), or measuring the deformation of a deformable object. Other methods are designed to work with non-rigid (bendable) objects, e.g. for posture independent shape retrieval (see for example
Spectral shape analysis Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami operator is invariant under isometries, it ...
).
Similarity classes
All
similar triangles have the same shape. These shapes can be classified using
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s ''u, v, w'' for the vertices, in a method advanced by J.A. Lester and
Rafael Artzy
Rafael Artzy (23 July 1912 – 22 August 2006) was an Israeli mathematician specializing in geometry.
Education and emigration
Artzy was born July 23, 1912, in Königsberg, Germany. His father was Edward I. Deutschlander and his mother Ida Freud ...
. For example, an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
can be expressed by the complex numbers 0, 1, (1 + i √3)/2 representing its vertices. Lester and Artzy call the ratio
:
the shape of triangle (''u, v, w''). Then the shape of the equilateral triangle is
:(0–(1+ √3)/2)/(0–1) = ( 1 + i √3)/2 = cos(60°) + i sin(60°) = exp( i π/3).
For any
affine transformation of the
complex plane,
a triangle is transformed but does not change its shape. Hence shape is an
invariant of
affine geometry
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle.
As the notion of '' parallel lines'' is one of the main properties that is ...
.
The shape ''p'' = S(''u,v,w'') depends on the order of the arguments of function S, but
permutations lead to related values. For instance,
:
Also
Combining these permutations gives
Furthermore,
:
These relations are "conversion rules" for shape of a triangle.
The shape of a
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
is associated with two complex numbers ''p,q''. If the quadrilateral has vertices ''u,v,w,x'', then ''p'' = S(''u,v,w'') and ''q'' = S(''v,w,x''). Artzy proves these propositions about quadrilateral shapes:
# If
then the quadrilateral is a
parallelogram.
# If a parallelogram has , arg ''p'' , = , arg ''q'' , , then it is a
rhombus
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
.
# When ''p'' = 1 + i and ''q'' = (1 + i)/2, then the quadrilateral is
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
.
# If
and sgn ''r'' = sgn(Im ''p''), then the quadrilateral is a
trapezoid
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium ().
A trapezoid is necessarily a convex quadrilateral in Eu ...
.
A
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 ...
has a shape defined by ''n'' – 2 complex numbers
The polygon bounds a
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
when all these shape components have imaginary components of the same sign.
Human perception of shapes
Human vision relies on a wide range of shape representations.
Some psychologists have theorized that humans mentally break down images into simple geometric shapes (e.g., cones and spheres) called
geons. Others have suggested shapes are decomposed into features or dimensions that describe the way shapes tend to vary, like their ''segmentability'', ''compactness'' and ''spikiness''. When comparing shape similarity, however, at least 22 independent dimensions are needed to account for the way natural shapes vary.
There is also clear evidence that shapes guide human
attention.
See also
*
Area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
*
Glossary of shapes with metaphorical names
Many shapes have metaphorical names, i.e., their names are metaphors: these shapes are named after a most common object that has it. For example, "U-shape" is a shape that resembles the letter U, a bell-shaped curve has the shape of the vertical ...
*
Lists of shapes
Lists of shapes cover different types of geometric shape and related topics. They include mathematics topics and other lists of shapes, such as shapes used by drawing or teaching tools.
Mathematics
* List of mathematical shapes
* List of two- ...
*
Shape factor
*
Size
Size in general is the magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to linear dimensions ( length, width, height, diameter, perimeter), area, or volume. Size can also be m ...
*
Solid geometry
In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), inc ...
*
Region (mathematics)
In mathematical analysis, a domain or region is a non-empty connected open set in a topological space, in particular any non-empty connected open subset of the real coordinate space or the complex coordinate space . This is a different concep ...
References
External links
* {{wiktionary-inline
Elementary geometry
Morphology
Structure