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Sphere with radius one, usually centered on the origin of the space
In mathematics, a unit sphere is a sphere of unit radius: the set of points at Euclidean distance 1 from some center point in three-dimensional space
Unit_sphere
Generalized sphere of dimension n (mathematics)
(n-1)} -sphere is the boundary of an n {\displaystyle n} -ball. Given a Cartesian coordinate system, the unit n {\displaystyle n} -sphere of radius
N-sphere
Set of points equidistant from a center
sphere: their length is twice the radius, d = 2r. Two points on the sphere connected by a diameter are antipodal points of each other. A unit sphere is
Sphere
Model of the extended complex plane plus a point at infinity
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the
Riemann_sphere
Study of angle-preserving transformations
general of orthogonal spheres, and in particular inversion in one of the spheres orthogonal to the unit sphere maps the unit sphere to itself. It also maps
Inversive_geometry
Representation of a quantum mechanical system
\mathbf {P} ^{1}.} This is the Bloch sphere, which can be mapped to the Riemann sphere. The Bloch sphere is a unit 2-sphere, with antipodal points corresponding
Bloch_sphere
Circle with radius of one
often denoted as S1 because it is a one-dimensional unit n-sphere. If (x, y) is a point on the unit circle's circumference, then |x| and |y| are the lengths
Unit_circle
Three-dimensional packing problem
Sphere packing in a sphere is a three-dimensional packing problem with the objective of packing a given number of equal spheres inside a unit sphere. It
Sphere_packing_in_a_sphere
Geometric concept
non-overlapping unit spheres (i.e., of radius 1) that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing
Kissing_number
Measure in 3-dimensional geometry
radian, sr = rad2. One steradian corresponds to one unit of area (of any shape) on the unit sphere surrounding the apex, so an object that blocks all rays
Solid_angle
Riemannian geometry, the unit tangent bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M), UTM, or SM is the unit sphere bundle for the tangent
Unit_tangent_bundle
Result on the topology of operators on an infinite-dimensional, complex Hilbert space
unit sphere. But the result was anyway essentially known (in 1935 Andrey Nikolayevich Tychonoff showed that the unit sphere was a retract of the unit
Kuiper's_theorem
Mathematical object
In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space
3-sphere
Geometry of figures on the surface of a sphere
denoted by the same upper case letters A, B, and C. Side lengths on a unit-radius sphere are denoted by lower-case letters: a, b, and c. The side lengths and
Spherical_trigonometry
SI derived unit of solid angle
angle subtended at the centre of a unit sphere by a unit area (of any shape) on its surface. For a general sphere of radius r, any portion of its surface
Steradian
Formula for the great-circle distance between two points on a sphere
versine of the angle θ, or the squares of half chord of the angle on a unit circle (sphere). It is related to other sinusoidal functions: hav θ = sin 2
Haversine_formula
Quadric surface that looks like a deformed sphere
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation
Ellipsoid
Property shared by codirectional lines
sphere and a ray in that direction emanating from the sphere's center; the tips of unit vectors emanating from a common origin point lie on the unit sphere
Direction_(geometry)
Affine connection on the tangent bundle of a manifold
unit circle. Let ⟨ , ⟩ be the usual scalar product on R3. Let S2 be the unit sphere in R3. The tangent space to S2 at a point m is naturally identified with
Levi-Civita_connection
Pathological embedding of the sphere in 3D space
The Alexander horned sphere is a pathological embedding of the 2-sphere into 3-dimensional Euclidean space. The topological object was discovered by J
Alexander_horned_sphere
Quaternion of norm 1 (unit quaternion)
unit. There is a sphere of imaginary units in the quaternions. Note that the expression for a versor is just Euler's formula for the imaginary unit
Versor
Coordinates comprising a distance and two angles
convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored, see graphic.) This (unit sphere) simplification
Spherical_coordinate_system
Set of points at distance less than one from a given point
upward onto the unit upper half-sphere, taking the "south-pole" of the unit sphere as the projection center, and then this half-sphere is projected sideways
Unit_disk
Theorem in probability theory
{\displaystyle X=(X_{1},\dots ,X_{d})} uniformly distributed on the unit sphere S d − 1 {\displaystyle S^{d-1}} , so that ‖ X ‖ = 1 {\displaystyle \|X\|=1}
Isserlis's_theorem
Inner product of a surface in 3D, induced by the dot product
^{2}}}\,du\,dv={\sqrt {EG-F^{2}}}\,du\,dv.} A spherical curve on the unit sphere in R3 may be parametrized as X ( u , v ) = [ cos u sin v sin u
First_fundamental_form
Spherical triangle with three right angles
coordinate system relative to which the sphere is a unit sphere. The spherical octant itself is the intersection of the sphere with one octant of space. Uniquely
Octant_of_a_sphere
Particular mapping that projects a sphere onto a plane
stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the pole or center of projection), onto a plane (the
Stereographic_projection
Topics referred to by the same term
Unit set, a singleton, a set with exactly 1 element Unit sphere, a sphere with a radius of length 1 Unit square, a square with sides of length 1 Unit
Unit
Differential geometry topic
map is a map N: X → S2 (where S2 is the unit sphere) such that for each p in X, the function value N(p) is a unit vector orthogonal to X at p. The Gauss
Gauss_map
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
distance of the points on the sphere from this origin can be assumed to be a unit length. With this convention, the n-sphere, S n {\displaystyle S^{n}}
Hopf_fibration
Property of all triangles on a Euclidean plane
the sides of the triangle. Because it is a unit sphere, a, b, and c are the angles at the center of the sphere subtended by those arcs, in radians. Let
Law_of_sines
How spheres of various dimensions can wrap around each other
structure of spheres viewed as topological spaces, forgetting about their precise geometry. The n-dimensional unit sphere — called the n-sphere for brevity
Homotopy_groups_of_spheres
Geometric theorem
earlier work by Giuseppe Vitali concerning the unit interval and on the paradoxical decompositions of the sphere by Felix Hausdorff, and discussed a number
Banach–Tarski_paradox
Special mathematical functions defined on the surface of a sphere
spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many
Spherical_harmonics
Continuous surjection satisfying a local triviality condition
tangent bundle T M {\displaystyle TM} , the unit sphere bundle is known as the unit tangent bundle. A sphere bundle is partially characterized by its Euler
Fiber_bundle
Topics referred to by the same term
Sphere (American band), American jazz ensemble Sphere (group), a Japanese pop idol unit Sphere (Polish band), a death metal band from Poland Sphere (album)
Sphere_(disambiguation)
Square with side length one
is a rational distance from all four vertices of the unit square. Unit circle Unit cube Unit sphere Horn, Alastair N. (1991), "IFSs and the Interactive
Unit_square
Mathematics lemma in functional analysis
vectors that are a distance of 1 {\displaystyle 1} from the origin the unit sphere, and denote the distance from a point u {\displaystyle u} to the set
Riesz's_lemma
Theorem in differential topology
even-dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in ℝ3 to every point p on a sphere such that
Hairy_ball_theorem
Statistical model used in machine learning
consider the normalized linear transform, which radially projects onto the unit sphere the output of an invertible linear transform, parametrized by the n -by-
Flow-based_generative_model
Generalized function whose value is zero everywhere except at zero
Laplacian to the integral with respect to the unit sphere measure dω of g(x · ξ) for ξ in the unit sphere Sn−1: δ ( x ) = Δ x ( n + k ) / 2 ∫ S n − 1 g
Dirac_delta_function
Geometrical object in four-dimensional space
torus; when a=b it is a square torus. If a2+b2=1, then Ta,b lies in the unit 3-sphere S3 ⊂ R4. The case a=b=1/√2 is a minimal surface in S3 and is often called
Clifford_torus
Simple curve of Euclidean geometry
centred at the origin with radius 1 is called the unit circle. Thought of as a great circle of the unit sphere, it becomes the Riemannian circle. Through any
Circle
Shortest distance between two points on the surface of a sphere
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 sphere. (By
Great-circle_distance
Basic circuit in quantum computing
⟩ {\displaystyle |\Psi \rangle } with n qubits is the surface of the unit sphere in C 2 n {\displaystyle \mathbb {C} ^{2^{n}}} and that the unitary transforms
Quantum_logic_gate
Recreational mathematics planar boundary and area problem
intersection body equals exactly half the volume of the unit sphere. The volume of the unit sphere reachable by the animal has the form of a three-dimensional
Goat_grazing_problem
configuration of n {\displaystyle n} mutually-repelling particles on a unit sphere? Convex uniform 5-polytopes – find and classify the complete set of these
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Electrical engineers graphical calculator
Riemann sphere. The mapping from the plane to the sphere is accomplished using a stereographic projection. The resulting Riemann sphere is a unit sphere with
Smith_chart
Entertainment venue in the Las Vegas Valley, United States
Sphere (also known as Sphere at the Venetian Resort or Las Vegas Sphere) is a music and entertainment arena in Paradise, Nevada, United States, east of
Sphere_(venue)
Japanese biological and chemical warfare unit (1936–1945)
Unit 731 (Japanese: 731部隊, Hepburn: Nana-san-ichi Butai), officially known as the Manchu Detachment 731 and also referred to as the Kamo Detachment and
Unit_731
Group of unitary complex matrices with determinant of 1
group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space
Special_unitary_group
Azimuthal equal-area map projection
on spheres of radius other than 1, using similar formulas. As defined in the preceding section, the Lambert azimuthal projection of the unit sphere is
Lambert azimuthal equal-area projection
Lambert_azimuthal_equal-area_projection
Cube with edge length one
between two random points in a unit cube Tychonoff cube, an infinite-dimensional analogue of the unit cube Unit square Unit sphere Ball, Keith (2010), "High-dimensional
Unit_cube
Diagrammatic representation of Sun's position over a period of time
the unit vector traces out 24 analemmas on the unit sphere centered on the chosen point. This unit sphere is equivalent to the celestial sphere. The
Analemma
Japanese voice actress and singer (born 1990)
Kannagi: Crazy Shrine Maidens. In 2009, she became part of the music unit Sphere, alongside Aki Toyosaki, Minako Kotobuki and Ayahi Takagaki. She released
Haruka_Tomatsu
Mathematics of smooth surfaces
The unit sphere in E3 has constant Gaussian curvature +1. The Euclidean plane and the cylinder both have constant Gaussian curvature 0. A unit pseudosphere
Differential geometry of surfaces
Differential_geometry_of_surfaces
Type of optimization problem
constraints cannot be eliminated by using generalized coordinates. Consider a unit sphere rolling without slipping on a plane. The configuration space of the system
Nonholonomic_system
Four-dimensional number system
and so cannot be a normed division algebra. The unit quaternions give a group structure on the 3-sphere S3 isomorphic to the groups Spin(3) and SU(2),
Quaternion
Theorem in quantum mechanics
frame function is a real-valued function f {\displaystyle f} on the unit sphere of a Hilbert space such that ∑ i f ( x i ) = 1 {\displaystyle \sum _{i}f(x_{i})=1}
Gleason's_theorem
Construct allowing differentiation of tangent vector fields of manifolds
metric. Let ⟨ , ⟩ be the usual scalar product on R3, and let S2 be the unit sphere. The tangent space to S2 at a point x is naturally identified with the
Affine_connection
American holding company in New York City
Sphere Entertainment Co. is an American entertainment holding company based in New York City, and controlled by the family of Charles Dolan. It owns the
Sphere_Entertainment
Mathematical idealization of the surface of a body
a polynomial, the surface is an algebraic surface. For example, the unit sphere is an algebraic surface, as it may be defined by the implicit equation
Surface_(mathematics)
Theorem in topology
the n {\displaystyle n} -dimensional sphere (or any symmetric domain that does not contain the origin). The unit circle is closed and bounded, but it
Brouwer_fixed-point_theorem
the unit sphere of the space. Equivalently, it is twice the infimum of distances between opposite points of the sphere, as measured within the sphere. Every
Girth_(functional_analysis)
Completion of the usual space with "points at infinity"
vector line intersects the unit sphere of V in two antipodal points, projective spaces can be equivalently defined as spheres in which antipodal points
Projective_space
chemistry and neutron transport. The surface integral of a function over the unit sphere, I [ f ] = ∫ d Ω f ( Ω ) = ∫ 0 π sin ( θ ) d θ ∫ 0 2 π d φ f ( θ ,
Lebedev_quadrature
Constant e raised to the power of pi
^{n}}{\Gamma (n+1)}}={\frac {\pi ^{n}}{n!}}.} Summing up all even-dimensional unit sphere volumes and utilizing the series expansion of the exponential function
Gelfond's_constant
Probability distribution on a sphere
and Christopher Bingham), is a probability distribution on the unit sphere (2-sphere S2 in 3-space R3). It is the analogue on S2 of the bivariate normal
Kent_distribution
Arrangement of points on a sphere
energy configuration of N electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist
Thomson_problem
Ugandan Social Media influencer / blogger born 1995 in mbarara town
closed minimal submanifolds M n {\displaystyle M^{n}} immersed in the unit sphere S n + m {\displaystyle S^{n+m}} with second fundamental form of constant
Chern's conjecture for hypersurfaces in spheres
Chern's_conjecture_for_hypersurfaces_in_spheres
Rational function of the form (az + b)/(cz + d)
inverse stereographic projection from the plane to the unit sphere, moving and rotating the sphere to a new location and orientation in space, and then
Möbius_transformation
Square roots of the eigenvalues of the self-adjoint operator
the singular values: Consider the image by T {\displaystyle T} of the unit sphere; this is an ellipsoid, and the lengths of its semi-axes are the singular
Singular_value
Distance from origin of tangent hyperplanes
convex set. Many authors restrict the support function to the Euclidean unit sphere and consider it as a function on Sn-1. The homogeneity property shows
Support_function
Matrix decomposition method
{\textstyle \mathbb {S} ^{n}} is the unit sphere in n dimensions. That is, the ellipsoid is a linear image of the unit sphere. Define the matrix V := [ v 1 |
Cholesky_decomposition
Assignment of a vector to each point in a subset of Euclidean space
(homeomorphic to the (n-1)-sphere) S around the zero, so that no other zeros lie in the interior of S. A map from this sphere to a unit sphere of dimension n − 1
Vector_field
Last letter of the Greek alphabet
surface S is defined as the surface area Ω of a unit sphere covered by the surface's projection onto the sphere. Braslavsky, S. E. (1 January 2007). "Glossary
Omega
Mathematical relation in spherical triangles
first law by polar duality. Let u, v, and w denote the unit vectors from the center of a unit sphere to those corners of the triangle. The angles and distances
Spherical_law_of_cosines
Mathematical entity to describe the probability of each possible measurement on a system
the set of all pure states corresponds to the unit sphere in the Hilbert space, because the unit sphere is defined as the set of all vectors with norm
Quantum_state
Scottish mathematician (1550–1617)
the centre of the sphere: on the unit sphere the side has length π/2. In the case that the side c has length π/2 on the unit sphere the equations governing
John_Napier
Mathematical theory
In stable homotopy theory, a branch of mathematics, the sphere spectrum S is the monoidal unit in the category of spectra. It is the suspension spectrum
Sphere_spectrum
Probability distribution in quantum mechanics
unit sphere for which polynomials of bounded degree can be averaged over to obtain the same value that integrating over surface measure on the sphere
Quantum_t-design
Constructing a strictly convex compact surface with specified Gaussian curvature
necessary and sufficient conditions on a non-negative Borel measure on the unit sphere Sn-1 to be the surface area measure of a convex body in R n {\displaystyle
Minkowski_problem
Unbounded quadric surface
a sphere: ... the equation of the unit sphere ρ2 + 1 = 0, and change the vector ρ to a bivector form, such as σ + τ √−1. The equation of the sphere then
Hyperboloid
Shortest network connecting points
bounded by the kissing number of tangent unit spheres. The total length of the edges, for points in a unit square, is at most proportional to the square
Euclidean minimum spanning tree
Euclidean_minimum_spanning_tree
World War II era Pan-Asian union under the Empire of Japan
The Greater East Asia Co-Prosperity Sphere (Japanese: 大東亜共栄圏, Hepburn: Dai Tōa Kyōeiken), also known as the GEACPS, was a pan-Asian union that the Empire
Greater East Asia Co-Prosperity Sphere
Greater_East_Asia_Co-Prosperity_Sphere
Formulas about vectors in three-dimensional Euclidean space
example, if four points are chosen on the unit sphere, A, B, C, D, and unit vectors drawn from the center of the sphere to the four points, a, b, c, d respectively
Vector_algebra_relations
Differential equation important in physics
on the unit sphere S, and ω is the area element on S. This result has the interpretation that u(t, x) is t times the mean value of φ on a sphere of radius
Wave_equation
Size of a mathematical ball
be expressed in terms of A n {\displaystyle A_{n}} , the area of the unit n-sphere. The first volumes are as follows: The n-dimensional volume of a Euclidean
Volume_of_an_n-ball
Artistic concept relating to perspective
Weiss "Vanishing Point Calculation as a Statistical Inference on the Unit Sphere" Proceedings of ICCV3, December, 1990 C. Coelho, M. Straforani, M. Campani
Vanishing_point
Result in integral geometry
unit sphere in R n | | unit ball in R n − 1 | = 2 π Γ ( n + 1 2 ) Γ ( n 2 ) {\displaystyle {\frac {|\partial S|}{E[|T(S)|]}}={\frac {|{\text{unit sphere
Crofton_formula
Italian mathematician (1835–1900)
curvature, the pseudosphere, and in the interior of an n-dimensional unit sphere, the so-called Beltrami–Klein model. He also developed singular value
Eugenio_Beltrami
Metric on a smooth statistical manifold
orthant (e.g. "quadrant" in R 2 {\displaystyle \mathbb {R} ^{2}} ) of a unit sphere, after appropriate changes of variable. Consider a flat, Euclidean space
Fisher_information_metric
Matrix decomposition
{\displaystyle \mathbf {D} \circ \mathbf {V} ^{*}} then sends the unit-sphere onto an ellipsoid isometric to T ( S ) {\displaystyle T(S)} . To define
Singular_value_decomposition
Theorem in physics
{\displaystyle \rho } . This mapping is continuous on the unit sphere of the Hilbert space, and since this unit sphere is connected, no continuous probability measure
Bell's_theorem
Free-flying robotic system
port. From 2006, three SPHERES units are being used in the International Space Station for a variety of experiments. The SPHERES Guest Scientist Program
SPHERES
Mapping which preserves all topological properties of a given space
Euclidean space. The stereographic projection is a homeomorphism between the unit sphere in R 3 {\displaystyle \mathbb {R} ^{3}} with a single point removed
Homeomorphism
Matrix representing a Euclidean rotation
(n + 1)-dimensional space will be a unit magnitude vector, which we may consider a point on a generalized sphere, Sn. Thus it is natural to describe the
Rotation_matrix
Theorem in geometry
Legendre, is stated as follows: Let ABC be a spherical triangle on the unit sphere with small sides a, b, c. Let A'B'C' be the planar triangle with the
Legendre's theorem on spherical triangles
Legendre's_theorem_on_spherical_triangles
Theorem in mathematics
the weak compactness of the unit sphere, Victor L. Klee reformulated this as a compactness criterion for the unit sphere in 1962 and assumes that this
James's_theorem
Formula in classical differential geometry
Clairaut, is a formula that characterizes the great circle paths on the unit sphere. The formula states that if γ {\displaystyle \gamma } is a parametrization
Clairaut's relation (differential geometry)
Clairaut's_relation_(differential_geometry)
UNIT SPHERE
UNIT SPHERE
Boy/Male
Hindu
Joyful unending, Calmness
Boy/Male
Celebrity, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Punjabi, Sanskrit, Sikh, Tamil, Telugu
Grown; Awakened; Shining
Girl/Female
American, British, English, Irish
Fair
Female
Hebrew
(×וּרִית) Hebrew name URIT means "fire, light."
Girl/Female
Hebrew
Light.
Boy/Male
Hindu
Knower of virtues, Talented, Excellent, Virtuous
Boy/Male
Bengali, English, Hindu, Indian
Dark Blue
Female
Egyptian
, Anahita ("pure, spotless").
Girl/Female
Hebrew
Graceful.
Boy/Male
Indian
Progress
Boy/Male
Indian
Unit of army
Boy/Male
Hindu
Pure or holy
Female
Welsh
Variant spelling of Welsh Enid, ENIT means "soul."
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Telugu
Holy; Untouched; Good; Pure
Girl/Female
Irish English
Together.
Female
English
English name derived from the vocabulary word, UNITY means "oneness, unity."
Male
English
Variant spelling of English Unni, UNI means "afflicted, depressed."
Boy/Male
Muslim
Unit of army
Boy/Male
Muslim/Islamic
Unit of army
Boy/Male
Indian
Who Won Every Time
UNIT SPHERE
UNIT SPHERE
Girl/Female
African, American, British, Chinese, Christian, Dutch, English, German, Greek, Hawaiian, Hebrew, Irish, Jamaican
Pure; Variation of Kay; Keeper of the Keys; Beloved; Crown of Laurels; Like God; Laurel; Crown; Slim; Fair
Boy/Male
Hindu, Indian
Power
Girl/Female
Arabic, Australian, Muslim
Brook; Rivulet; Small Stream
Boy/Male
Muslim
Prosperity. Populousness.
Girl/Female
Hebrew
Genisia, the Virgin Mary of Turin, is a protectress invoked against drought in Catholic tradition.
Boy/Male
Hindu, Indian
Lord Shiva
Boy/Male
Indian, Punjabi, Sikh
Light of Shelter
Boy/Male
Hindu
Intelligent, Thoughtful
Girl/Female
Indian
Unique
Girl/Female
Hindu
UNIT SPHERE
UNIT SPHERE
UNIT SPHERE
UNIT SPHERE
UNIT SPHERE
n.
Concord; harmony; conjunction; agreement; uniformity; as, a unity of proofs; unity of doctrine.
v. t.
To knit or bind together; to unite closely.
n.
A single thing, as a magnitude or number, regarded as an undivided whole.
v. t.
United; joint; as, unite consent.
n.
The number greater by a unit than seventeen; eighteen units or objects.
imp. & p. p.
of Knit
v. i.
To be united closely; to grow together; as, broken bones will in time knit and become sound.
v. t.
To put together so as to make one; to join, as two or more constituents, to form a whole; to combine; to connect; to join; to cause to adhere; as, to unite bricks by mortar; to unite iron bars by welding; to unite two armies.
v. t.
To unite closely; to knit together.
a.
Of or pertaining to a unit or units; relating to unity; as, the unitary method in arithmetic.
n.
The number greater by a unit than two; three units or objects.
v. t.
To form, as a textile fabric, by the interlacing of yarn or thread in a series of connected loops, by means of needles, either by hand or by machinery; as, to knit stockings.
n.
Any one of numerous species of fresh-water mussels belonging to Unio and many allied genera.
v. t.
To unite closely; to connect; to engage; as, hearts knit together in love.
v. t.
To knit together; to unite closely; to intertwine.
v. t.
To unite.
v. t.
To remove the turns of (a rope or cable) from the bits; as, to unbit a cable.
n.
The number greater by a unit than seven; eight units or objects.
n.
Any definite quantity, or aggregate of quantities or magnitudes taken as one, or for which 1 is made to stand in calculation; thus, in a table of natural sines, the radius of the circle is regarded as unity.
n.
The number greater than eight by a unit; nine units or objects.