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Graphical notation for multilinear algebra calculations
In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions
Penrose_graphical_notation
Algebraic object with geometric applications
distinct pairs of indices may be summed this way. Penrose graphical notation is a diagrammatic notation which replaces the symbols for tensors with shapes
Tensor
Topics referred to by the same term
notation (dance) A diagrammatic notation in mathematical notation In physics: Penrose graphical notation Coxeter–Dynkin diagram A visual programming language
Graphic_notation
Graphical representation of a morphism
tensor product, string diagrams are called tensor networks or Penrose graphical notation. This has led to the development of categorical quantum mechanics
String_diagram
British theoretical physicist (born 1929)
Oliver Penrose FRS FRSE (born 6 June 1929) is a British theoretical physicist and emeritus professor at Heriot-Watt University. His topics of interest
Oliver_Penrose
Mathematical notation for tensors and spinors
_{3}}\omega _{\sigma (a)\sigma (b)\sigma (c)}} Penrose graphical notation Einstein notation Index notation Tensor Antisymmetric tensor Raising and lowering
Abstract_index_notation
Model of quantum computing
physical cables. The graphical depiction of quantum circuit elements is described using a variant of the Penrose graphical notation.[citation needed] Richard
Quantum_circuit
Graphical language for quantum processes
These are connected together to form a tensor network similar to Penrose graphical notation. Due to the symmetries of the spiders and the properties of the
ZX-calculus
Mathematical study of illumination of rooms with mirrored walls
The original problem was first solved in 1958 by Roger Penrose using ellipses to form the Penrose unilluminable room. He showed that there exists a room
Illumination_problem
British geneticist
Shirley Victoria Penrose Hodgson (born 22 February 1945) is a British geneticist. Hodgson studied at Somerville College, Oxford. She worked as a GP, then
Shirley_Hodgson
Diagram used to represent quantum field theory calculations
representations of matrix groups. The diagrammatic notation can thus greatly simplify calculations. Roger Penrose described spin networks in 1971. Spin networks
Spin_network
Shorthand notation for tensor operations
\alpha }} Tensor Abstract index notation Bra–ket notation Penrose graphical notation Levi-Civita symbol DeWitt notation This applies only for numerical
Einstein_notation
Tensor index notation for tensor-based calculations
Metric tensor Multilinear algebra Multilinear subspace learning Penrose graphical notation Regge calculus Ricci calculus Ricci decomposition Tensor (intrinsic
Ricci_calculus
Claim that human mathematicians are not describable as formal proof systems
The Penrose–Lucas argument is a logical argument partially based on Kurt Gödel's first incompleteness theorem. In 1931, Gödel proved that every effectively
Penrose–Lucas_argument
System of symbolic representation
mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are Penrose graphical notation and Coxeter–Dynkin
Mathematical_notation
Origin and evolution of the symbols used to write equations and formulas
fields called a tetrad). In the 1990s, Roger Penrose proposed Penrose graphical notation (tensor diagram notation) as a, usually handwritten, visual depiction
History of mathematical notation
History_of_mathematical_notation
Roger Penrose: Moore–Penrose inverse, the most widely known generalization of the inverse matrix in particular linear algebra Penrose graphical notation, a
List of things named after Roger Penrose
List_of_things_named_after_Roger_Penrose
Mathematical Concept
associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas
Voigt_notation
Branch of mathematics
tensors Dyadic tensor Glossary of tensor theory Metric tensor Bra–ket notation Multilinear subspace learning Multivector Geometric algebra Clifford algebra
Multilinear_algebra
Tensor equal to the negative of any of its transpositions
Vectors to Tensors. Springer. p. 225. ISBN 978-3-540-22887-5. section §7. Penrose, Roger (2007). The Road to Reality. Vintage books. ISBN 978-0-679-77631-4
Antisymmetric_tensor
Effect in special relativity
also sometimes referred to as the Penrose–Terrell effect, the Terrell–Penrose effect or the Lampa–Terrell–Penrose effect. In 1924, a paper by Anton Lampa
Terrell_rotation
Array of numbers
or no columns, called an empty matrix. The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are commonly written
Matrix_(mathematics)
Quantum state of multiple particles represented as complex matrices
and. In the context of finite automata see. For emphasis placed on the graphical reasoning of tensor networks, see the introduction. For a system of N
Matrix_product_state
Philosophical argument based on the theory of relativity
Putnam (1967). It is sometimes called the Rietdijk–Putnam–Penrose argument. Roger Penrose advanced a form of this argument that has been called the Andromeda
Rietdijk–Putnam_argument
English mathematician, mathematical physicist (born 1931)
Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, and philosopher of science. He is Emeritus Rouse Ball Professor
Roger_Penrose
contrast, a dyad is specifically a dyadic tensor of rank one. Einstein notation This notation is based on the understanding that whenever a multidimensional array
Glossary_of_tensor_theory
Mathematical notation
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory
Multi-index_notation
Matrix operation which flips a matrix over its diagonal
another matrix, called the transpose of A and often denoted AT (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician
Transpose
Algebraic operation on coordinate vectors
specified with respect to an orthonormal basis, is defined, in summation notation, as: a ⋅ b = ∑ i = 1 n a i b i = a 1 b 1 + a 2 b 2 + ⋯ + a n b n {\displaystyle
Dot_product
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
developing Aitken's diagrams, to become part of the technique of Penrose graphical notation. Also, this relation is extensively used in S-duality theories
Kronecker_delta
Pictorial computational technique in quantum chemistry
notation and include the abstract nature of the state, such as tensor products and transformation rules. The notation parallels the idea of Penrose graphical
Angular momentum diagrams (quantum mechanics)
Angular_momentum_diagrams_(quantum_mechanics)
Quantum mechanics posed in terms of category theory
diagrams. These diagrammatic languages can be traced back to Penrose graphical notation, developed in the early 1970s. Diagrammatic reasoning has been
Categorical_quantum_mechanics
Specialized notation for multivariable calculus
calculus. Supports general symbolic tensor derivatives using Penrose graphical notation. Matrix Reference Manual, Mike Brookes, Imperial College London
Matrix_calculus
Differential form of degree one or section of a cotangent bundle
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
One-form
Belgian theoretical physicist and logician
the development of a diagrammatic quantum formalism based on Penrose graphical notation, on which he wrote a textbook entitled Picturing Quantum Processes
Bob_Coecke
1989 book by Roger Penrose
is a 1989 book by the mathematical physicist Roger Penrose that posits a quantum mind theory. Penrose argues that human consciousness is non-algorithmic
The_Emperor's_New_Mind
Algebra associated to any vector space
algebra, a quantum deformation of the symmetric algebra by a symplectic form Penrose, R. (2007). The Road to Reality. Vintage books. ISBN 978-0-679-77631-4
Exterior_algebra
Tensor that describes the 4D geometry of spacetime
{\displaystyle g_{\mu \nu }} themselves as the metric (see, however, abstract index notation). With the quantities d x μ {\displaystyle dx^{\mu }} being regarded as
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Expression that may be integrated over a region
dependent is zero. A common notation for the wedge product of elementary k {\displaystyle k} -forms is so called multi-index notation: in an n {\displaystyle
Differential_form
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
\end{aligned}}} It is common in rigid body mechanics to use notation that explicitly identifies the x {\displaystyle x} , y {\displaystyle y}
Moment_of_inertia
Graphical means of performing computations in linear algebra
have simple diagrammatic proofs. They are closely related to Penrose's graphical notation. Let V be a vector space of dimension n over a field F (with
Trace_diagram
Specification of a derivative along a tangent vector of a manifold
language and using a local coordinate system and the traditional index notation. The covariant derivative of a tensor field is presented as an extension
Covariant_derivative
Vector behavior under coordinate changes
opposed to those of covectors) are said to be contravariant. In Einstein notation (implicit summation over repeated index), contravariant components are
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Conserved physical quantity; rotational analogue of linear momentum
about the center of rotation – circular, linear, or otherwise. In vector notation, the orbital angular momentum of a point particle in motion about the origin
Angular_momentum
Tensor having both covariant and contravariant indices
covariant, the last one contravariant, and the remaining ones mixed. Notationally, these tensors differ from each other by the covariance/contravariance
Mixed_tensor
Mathematical operation on vector spaces
differentiable, then a */ b is differentiable. However, these kinds of notation are not universally present in array languages. Other array languages may
Tensor_product
Theory of gravitation as curved spacetime
introduction to the necessary mathematics Poisson 2004. For the Penrose process, see Penrose 1969 Bekenstein 1973, Bekenstein 1974 The fact that black holes
General_relativity
Decomposition in multilinear algebra
{\displaystyle M>2} and all I m ≥ 2 {\displaystyle I_{m}\geq 2} . For simplicity in notation, assume without loss of generality that the factors are ordered such that
Tensor_rank_decomposition
Topological space that locally resembles Euclidean space
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Manifold
Operation in mathematics
2x2; often 3x3 or 4x4 are used, but any size is allowed. In simple index notation, this is written ∑ j = 1 2 a i j × b j k = c i k {\textstyle \sum _{j=1}^{2}a_{ij}\times
Tensor_contraction
Abbreviation in the fields of special and general relativity
four-dimensional spacetime. General four-tensors are usually written in tensor index notation as A ν 1 , ν 2 , . . . , ν m μ 1 , μ 2 , . . . , μ n {\displaystyle A_{\;\nu
Four-tensor
Method for specifying point positions
coordinates Frame of reference Galilean transformation Grid reference Nomogram, graphical representations of different coordinate systems Reference system Rotation
Coordinate_system
Exterior algebraic map taking tensors from p forms to n-p forms
}(dy\wedge dz)&=dt\wedge dx\,.\end{aligned}}} These are summarized in the index notation as ⋆ ( d x μ ) = η μ λ ε λ ν ρ σ 1 3 ! d x ν ∧ d x ρ ∧ d x σ , ⋆ ( d x
Hodge_star_operator
Theory of interwoven space and time by Albert Einstein
would be observed as length contracted. In 1959, James Terrell and Roger Penrose independently pointed out that differential time lag effects in signals
Special_relativity
Isomorphism between the tangent and cotangent bundles of a manifold
the use of the musical notation symbols ♭ {\displaystyle \flat } (flat) and ♯ {\displaystyle \sharp } (sharp). In the notation of Ricci calculus and mathematical
Musical_isomorphism
Affine connection on the tangent bundle of a manifold
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Levi-Civita_connection
Operation that pairs a left and a right R-module into an abelian group
_{R}N} . It is often called a pure tensor. Strictly speaking, the correct notation would be x ⊗R y but it is conventional to drop R here. Then, immediately
Tensor_product_of_modules
Tensor in differential geometry
v 1 , … , v n {\displaystyle v_{1},\ldots ,v_{n}} . In abstract index notation, R i c a b = R c b c a = R c a c b . {\displaystyle \mathrm {Ric} _{ab}=\mathrm
Ricci_curvature
Physical phenomenon
Bibcode:2010ConPh..51...59C. doi:10.1080/00107510903257624. S2CID 752173. R. Penrose, Applications of negative dimensional tensors, In: Combinatorial Mathematics
Quantum_teleportation
Physics concept
{x}^{i}}}} This is the explicit form of the covariant transformation rule. The notation of a normal derivative with respect to the coordinates sometimes uses a
Covariant_transformation
Class of mathematical software
manipulation. Supports general symbolic tensor derivatives using Penrose graphical notation, and gaussian expectations via Isserlis' theorem. TensorDecompositions
Tensor_software
Structure defining distance on a manifold
is increased by du units, and v is increased by dv units. Using matrix notation, the first fundamental form becomes d s 2 = [ d u d v ] [ E F F G ] [ d
Metric_tensor
Notation used for Weyl spinors
In theoretical physics, Van der Waerden notation refers to the usage of two-component spinors (Weyl spinors) in four spacetime dimensions. This is standard
Van_der_Waerden_notation
Tensor invariant under permutations of vectors it acts on
the operator is omitted: T1T2 = T1 ⊙ T2. In some cases an exponential notation is used: v ⊙ k = v ⊙ v ⊙ ⋯ ⊙ v ⏟ k times = v ⊗ v ⊗ ⋯ ⊗ v ⏟ k times =
Symmetric_tensor
Coordinate-free definition of a tensor
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Tensor_(intrinsic_definition)
Function that is invariant under all permutations of its variables
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Symmetric_function
Construct in differenital geometry
{\displaystyle A_{j}{}^{k}\ =\ \Gamma ^{k}{}_{ij}\,dx^{i}.} The point of the notation is to distinguish the indices j, k, which run over the n dimensions of
Metric_connection
General relativity articles using tensors will use the abstract index notation. The principle of general covariance was one of the central principles
Mathematics of general relativity
Mathematics_of_general_relativity
Mathematical function, in linear algebra
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Linear_map
Study of curves from a differential point of view
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Differentiable_curve
Branch of mathematics
popularised the tensor calculus of Ricci and Levi-Civita and introduced the notation g {\displaystyle g} for a Riemannian metric, and Γ {\displaystyle \Gamma
Differential_geometry
Non-tensorial representation of the spin group
in Mathematics. 14: 1–55. doi:10.1016/0001-8708(74)90021-8. MR 0358873. Penrose, Roger; Rindler, W. (1988). Spinor and twistor methods in space-time geometry
Spinor
Property of a mathematical space
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Dimension
Mapping from p forms to p-1 forms
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Interior_product
Set of vectors used to define coordinates
j}y_{j},} for i = 1, ..., n. This formula may be concisely written in matrix notation. Let A be the matrix of the a i , j {\displaystyle a_{i,j}} , and X = [
Basis_(linear_algebra)
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Symmetrization
Representation of mechanical stress at every point within a deformed 3D object
tensor transformation law under a change in the system of coordinates. A graphical representation of two-dimensional coordinate transformations Mohr's circle
Cauchy_stress_tensor
Straight path on a curved surface or a Riemannian manifold
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Geodesic
Assignment of a tensor continuously varying across a region of space
curvature tensors built from them are. The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent
Tensor_field
Generalization of tensor fields
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Tensor_density
System of moving vectors in differential geometry
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Parallel_transport
Antisymmetric permutation object acting on tensors
lower case epsilon ε or ϵ, or less commonly the Latin lower case e. Index notation allows one to display permutations in a way compatible with tensor analysis:
Levi-Civita_symbol
Element of an exterior algebra
image analysis and applications. Springer. p. 25. ISBN 3-540-23527-2. R. Penrose (2007). The Road to Reality. Vintage books. ISBN 978-0-679-77631-4. J.A
Multivector
Differential form
{\displaystyle \omega } is frequently used to denote the volume form, this notation is not universal; the symbol ω {\displaystyle \omega } often carries many
Volume_form
Concept in differential geometry
form ψ is horizontal if ψ(v0, ..., vk) = ψ(hv0, ..., hvk).) By abuse of notation, the differential of ρ at the identity element may again be denoted by
Exterior_covariant_derivative
Continuous surjection satisfying a local triviality condition
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Fiber_bundle
Tensor field in Riemannian geometry
noncommutativity of the second covariant derivative. In abstract index notation, R d c a b Z c = ∇ a ∇ b Z d − ∇ b ∇ a Z d . {\displaystyle R^{d}{}_{cab}Z^{c}=\nabla
Riemann_curvature_tensor
Object in differential geometry
part and another part which contains the trace terms. Using the index notation, the trace of T is given by a i = T k i k , {\displaystyle a_{i}=T^{k}{}_{ik}
Torsion_tensor
Basis used to express spherical tensors
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Spherical_basis
Tensor operator generalizes the notion of operators which are scalars and vectors
Italiana di Fisica, IOS. ISBN 978-905-199-24-72. Introduction to the Graphical Theory of Angular Momentum. Springer. 2009. ISBN 978-364-203-11-99. A
Tensor_operator
discovered – Joseph Priestley Pell's equation – John Pell Penrose graphical notation – Roger Penrose Periodic Table – John Alexander Reina Newlands pion and
List of British innovations and discoveries
List_of_British_innovations_and_discoveries
Electromagnetism in general relativity
square brackets indicate anti-symmetrization (see Ricci calculus for the notation). The covariant derivative of the electromagnetic field is F α β ; γ =
Maxwell's equations in curved spacetime
Maxwell's_equations_in_curved_spacetime
Math/physics concept
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Connection_form
Type of physical quantity
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Pseudotensor
Measure of the curvature of a pseudo-Riemannian manifold
v_{3}\right)k\left(v_{1},v_{4}\right)\end{aligned}}} In tensor component notation, this can be written as C i k ℓ m = R i k ℓ m + 1 n − 2 ( R i m g k ℓ −
Weyl_tensor
become smaller: 1 Kelvin per m becomes 0.001 Kelvin per mm. In Einstein notation, contravariant vectors and components of tensors are shown with superscripts
Introduction to the mathematics of general relativity
Introduction_to_the_mathematics_of_general_relativity
Branch of physics which studies the behavior of materials modeled as continuous media
Transport phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad
Continuum_mechanics
British physiologist and biochemist (1864–1956)
Leathes married Lionel Penrose MD, FRS, Professor of Genetics at University College, London in 1928, and was the mother of Oliver Penrose, the theoretical physicist;
John_Beresford_Leathes
Type of derivative in differential geometry
=f{\mathcal {L}}_{X}\omega +df\wedge i_{X}\omega .} In local coordinate notation, for a type ( r , s ) {\displaystyle (r,s)} tensor field T {\displaystyle
Lie_derivative
Array of numbers describing a metric connection
reminder that these are defined to be equivalent notation for the same concept. The choice of notation is according to style and taste, and varies from
Christoffel_symbols
PENROSE GRAPHICAL-NOTATION
PENROSE GRAPHICAL-NOTATION
Surname or Lastname
English
English : variant of Pearce.
Male
Welsh
Welsh form of Greek Petros, PEDR means "rock, stone."
Male
Irish
Irish Gaelic form of Greek Petros, PIARAS means "rock, stone."
Boy/Male
Italian Spanish
Enduring. The poet Dante Alighieri wrote The Divine Comedy with its graphic description of...
Boy/Male
Irish
Comes from the Norman French name “â€Piersâ€â€ and is still very popular as it is given to honor Patrick Pearse, one of the leaders of the Easter Rising of 1916 when Ireland won its independence from England.
Male
Finnish
Finnish form of Greek Petros, PEKKA means "rock, stone."
Boy/Male
Hindu, Indian
Prose
Boy/Male
Italian Spanish
Enduring. The poet Dante Alighieri wrote The Divine Comedy with its graphic description of...
Boy/Male
Australian, British, English, Irish
From the Piers; Tone; Rock
Male
Finnish
Finnish form of Greek Petros, PIETARI means "rock, stone."
Male
Romanian
Romanian form of Greek Petros, PETRE means "rock, stone."
Boy/Male
German
Famous Commander
Girl/Female
Arabic
Fragments; Prose Writer
Boy/Male
Tamil
Prose
Boy/Male
Spanish American Italian Latin
Enduring. The poet Dante Alighieri wrote The Divine Comedy with its graphic description of...
Boy/Male
Arabic, Muslim
Prose Writer
Boy/Male
Australian, Greek
A Rock; Form of Peter
Male
Greek
Greek translation of the Aramaic byname Kephas, PETROS means "rock, stone." In the bible, this is the name of one of Christ's apostles. The name was given by Jesus to Simon son of Jona, to distinguish him from Simon Zelotes.Â
Boy/Male
Italian Spanish
Enduring. The poet Dante Alighieri wrote The Divine Comedy with its graphic description of...
Male
Polish
Polish form of Greek Petros, PIOTR means "rock, stone."
PENROSE GRAPHICAL-NOTATION
PENROSE GRAPHICAL-NOTATION
Boy/Male
Hindu, Indian, Tamil
Murugan
Girl/Female
Arabic
Pretty; Beautiful
Girl/Female
Hindu, Indian, Tamil
Rainfall
Girl/Female
Australian, Chinese, French, German, Italian, Latin, Portuguese, Swiss
Bean Farmer; Form of Fabian; Bean Grower; A Bean; Female Version of Fabio
Girl/Female
Teutonic
Wealthy.
Female
Danish
, pearl.
Male
Native American
Native American Cheyenne name TAHMELAPACHME means "dull knife."
Girl/Female
Hindu
A Goddess
Girl/Female
Indian
Fragrance
Boy/Male
Tamil
Fragrance, Aroma
PENROSE GRAPHICAL-NOTATION
PENROSE GRAPHICAL-NOTATION
PENROSE GRAPHICAL-NOTATION
PENROSE GRAPHICAL-NOTATION
PENROSE GRAPHICAL-NOTATION
a.
Operose.
a.
Of or pertaining to a seraph; becoming, or suitable to, a seraph; angelic; sublime; pure; refined.
a.
Of or pertaining to the arts of painting and drawing.
a.
Having the faculty of, or characterized by, clear and impressive description; vivid; as, a graphic writer.
a.
Possessing or exhibiting unpoetical characteristics; plain; dull; prosaic; as, the prose duties of life.
imp. & p. p.
of Peruse
a.
Leprose.
p. pr. & vb. n.
of Peruse
a.
Written or engraved; formed of letters or lines.
adv.
In a graphic manner; vividly.
v. t.
To reduce to prose.
v. i.
To write prose.
a.
Having numerous or conspicuous veins; veiny; as, a venose frond.
a.
Pertaining to, or composed of, prose; not in verse; as, prose composition.
a.
Of or pertaining to the art of writing.
a.
Alt. of Seraphical
v. t.
To write in prose.
n.
Graphic granite. See under Granite.
a.
Well delineated; clearly and vividly described.
a.
Alt. of Graphical