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# Classifications of figurate numbers

With the exception of the hyperpyramidal numbers (which include the pyramidal numbers and the centered pyramidal numbers, i.e. the (centered polygons) pyramidal numbers, as 3-dimensional hyperpyramidal numbers), all the figurate numbers considered are regular polytope numbers corresponding to regular convex polytopes in a
 d
-dimensional Euclidean space
 ℝ d, d   ≥   0.
Among the hyperpyramidal numbers, the
 d
-dimensional square hyperpyramidal numbers, although not corresponding to regular polytopes, are of particular interest since they are building blocks for the construction of the hyperoctahedral numbers (orthoplicial polytopic numbers), which are regular polytopes. For example, the
 n
th octahedral number is the
 n
th square dipyramidal number, i.e. it is the adjunction of the
 n
th square pyramidal number to the
 (n  −  1)
th square pyramidal number (corresponding to joined square pyramids at their square bases), while for hyperoctahedral numbers of dimension
 d   ≥   3
we must do
 d  −  2

Otherwise, considering nonconvex regular (e.g. stellated) polytopic numbers or considering nonregular (e.g. Archimedean solids) polytopic numbers would open the door to a humongous number of possibilities...

Category:Figurate numbers (by increasing number of vertices)
Figurate numbers
Category:Classifications of figurate numbers
Classifications of figurate numbers
Category:Regular polytope numbers (by increasing number of vertices)
Regular polytope numbers
Category:Simplicial polytopic numbers
Simplicial polytopic numbers or Regular simplex numbers or Simplex numbers or
 αd
numbers (dimensionality
 d   ≥   0
)
Triangular numbers (3-sided Polygonal numbers or 2-D Simplicial polytopic numbers)
Tetrahedral numbers (4-faced Platonic numbers or 3-D Simplicial polytopic numbers)
Hypertetrahedral numbers (REDIRECT to Simplicial polytopic numbers)
Category:Centered simplicial polytopic numbers
Centered simplicial polytopic numbers or Centered regular simplex numbers or Centered simplex numbers or centered
 αd
numbers (dimensionality
 d   ≥   0
)
Centered triangular numbers (3-sided Centered polygonal numbers or 2-D Centered simplicial polytopic numbers)
Centered tetrahedral numbers (4-faced Centered Platonic numbers or 3-D Centered simplicial polytopic numbers)
Centered hypertetrahedral numbers (REDIRECT to Centered simplicial polytopic numbers)
Category:Orthoplicial polytopic numbers
Orthoplicial polytopic numbers or Orthoplex numbers or Cross polytope numbers or
 βd
numbers (dimensionality
 d   ≥   0
)
Square numbers (4-sided Polygonal numbers, 2-D Regular orthotopic numbers or 2-D Orthoplicial polytopic numbers)
Octahedral numbers or Square dipyramidal numbers (8-faced Platonic numbers or 3-D Orthoplicial polytopic numbers)
Hyperoctahedral numbers (REDIRECT to Orthoplicial polytopic numbers)
Category:Centered orthoplicial polytopic numbers
Centered orthoplicial polytopic numbers or Centered orthoplex numbers or Centered cross polytope numbers or centered
 βd
numbers (dimensionality
 d   ≥   0
)
Centered square numbers (4-sided Centered polygonal numbers, 2-D Centered regular orthotopic numbers or 2-D Centered orthoplicial polytopic numbers)
Centered octahedral numbers or (Centered squares) dipyramidal numbers (8-faced Centered Platonic numbers or 3-D Centered orthoplicial polytopic numbers)
Centered hyperoctahedral numbers (REDIRECT to Centered orthoplicial polytopic numbers)
Category:Regular orthotopic numbers
Regular orthotopic numbers or Orthotope numbers or Measure polytope numbers or
 γd
numbers (dimensionality
 d   ≥   0
)
Square numbers (4-sided Polygonal numbers, 2-D Regular orthotopic numbers or 2-D Orthoplicial polytopic numbers)
Cube numbers (6-faced Platonic numbers or 3-D Regular orthotopic numbers)
Hypercube numbers (REDIRECT to Regular orthotopic numbers)
Category:Centered regular orthotopic numbers
Centered regular orthotopic numbers or Centered orthotope numbers or Centered measure polytope numbers or centered
 γd
numbers (dimensionality
 d   ≥   0
)
Centered square numbers (4-sided Centered polygonal numbers, 2-D Centered regular orthotopic numbers or 2-D Centered orthoplicial polytopic numbers)
Centered cube numbers (6-faced Centered Platonic numbers or 3-D Centered regular orthotopic numbers)
Centered hypercube numbers (REDIRECT to Centered regular orthotopic numbers)
Category:Nonregular polytope numbers (by increasing number of vertices)
Nonregular polytope numbers (created as stub)
Category:Hyperpyramidal numbers
 Category:(Centered squares) hyperpyramidal numbers Category:Centered square hyperpyramidal numbers REDIRECT to Category:(Centered squares) hyperpyramidal numbers NOTE: The (centered squares) hyperpyramidal numbers ARE NOT GLOBALLY CENTERED (only the original squares are) (Centered squares) hyperpyramidal numbers Centered square hyperpyramidal numbers REDIRECT to (Centered squares) hyperpyramidal numbers
Category:3-dimensional hyperpyramidal numbers
 Category:(Centered polygons) pyramidal numbers Category:Centered pyramidal numbers (REDIRECT to Category:(Centered polygons) pyramidal numbers) NOTE: The (centered polygons) pyramidal numbers ARE NOT GLOBALLY CENTERED (only the original polygons are) (Centered polygons) pyramidal numbers Centered pyramidal numbers (REDIRECT to (Centered polygons) pyramidal numbers)
Category:Classification of figurate numbers by dimensionality
Classification of figurate numbers by dimensionality
Category:0-dimensional figurate numbers (by increasing number of vertices)
Category:0-dimensional regular polytope numbers
Category:Point numbers
Point numbers {0, 1} (The convention being that figurate numbers have the initial dot for
 n = 1
.)
0-dimensional regular polytope numbers (REDIRECT to Point numbers)
Category:Centered point numbers
Centered point numbers {1} (The convention being that centered figurate numbers have the central dot for
 n = 0
.)
Centered 0-dimensional regular polytope numbers (REDIRECT to Centered point numbers)
 Category:1-dimensional figurate numbers (by increasing number of vertices)
Category:2-dimensional figurate numbers (by increasing number of vertices)
Category:2-dimensional regular polytope numbers
 Category:Polygonal numbers Polygonal numbers 2-dimensional regular polytope numbers (REDIRECT to Polygonal numbers) Triangular numbers (3-sided Polygonal numbers or 2-D Simplicial polytopic numbers) Square numbers (4-sided Polygonal numbers, 2-D Regular orthotopic numbers or 2-D Orthoplicial polytopic numbers) Pentagonal numbers (5-sided Polygonal numbers) Hexagonal numbers (6-sided Polygonal numbers) Heptagonal numbers (7-sided Polygonal numbers) Octagonal numbers (8-sided Polygonal numbers) Dodecagonal numbers (12-sided Polygonal numbers) Icosagonal numbers (20-sided Polygonal numbers)
 Category:Centered polygonal numbers Centered polygonal numbers Centered 2-dimensional regular polytope numbers (REDIRECT to Centered polygonal numbers) Centered triangular numbers (3-sided Centered polygonal numbers or 2-D Centered simplicial polytopic numbers) Centered square numbers (4-sided Centered polygonal numbers, 2-D Centered regular orthotopic numbers or 2-D Centered orthoplicial polytopic numbers) Centered pentagonal numbers (5-sided Centered polygonal numbers) Centered hexagonal numbers or Hex numbers (6-sided Centered polygonal numbers) Centered heptagonal numbers (7-sided Centered polygonal numbers) Centered octagonal numbers (8-sided Centered polygonal numbers) Centered dodecagonal numbers (12-sided Centered polygonal numbers) Centered icosagonal numbers (20-sided Centered polygonal numbers)
Category:3-dimensional figurate numbers (by increasing number of vertices)
Category:3-dimensional regular polytope numbers
Category:Platonic numbers
Platonic numbers
3-dimensional regular polytope numbers (REDIRECT to Platonic numbers)
 Tetrahedral numbers (4-faced Platonic numbers or 3-D Simplicial polytopic numbers)
 Octahedral/cube numbers (dual pair) Octahedral numbers or Square dipyramidal numbers (8-faced Platonic numbers or 3-D Orthoplicial polytopic numbers) Cube numbers (6-faced Platonic numbers or 3-D Regular orthotopic numbers)
 Icosahedral/dodecahedral numbers (dual pair) Icosahedral numbers (REDIRECT to 20-faced Platonic numbers) Dodecahedral numbers (REDIRECT to 12-faced Platonic numbers)
Category:Centered Platonic numbers
Centered Platonic numbers
Centered 3-dimensional regular polytope numbers (REDIRECT to Centered Platonic numbers)
 Centered octahedral/cube numbers (dual pair) Centered octahedral numbers or (Centered squares) dipyramidal numbers or (8-faced Centered Platonic numbers or 3-D Centered orthoplicial polytopic numbers) Centered cube numbers (6-faced Centered Platonic numbers or 3-D Centered regular orthotopic numbers)
 Centered icosahedral/dodecahedral numbers (dual pair) Centered icosahedral numbers (REDIRECT to 20-faced Centered Platonic numbers) Centered dodecahedral numbers (REDIRECT to 12-faced Centered Platonic numbers)
Category:3-dimensional nonregular polytope numbers
 Category:(Centered polygons) pyramidal numbers Category:Centered pyramidal numbers (REDIRECT to Category:(Centered polygons) pyramidal numbers) NOTE: The (centered polygons) pyramidal numbers ARE NOT GLOBALLY CENTERED (only the original polygons are) (Centered polygons) pyramidal numbers Centered pyramidal numbers (REDIRECT to (Centered polygons) pyramidal numbers)
 Dipyramidal/prism numbers (dual pair)
 Centered dipyramidal/prism numbers (dual pair)
Category:4-dimensional figurate numbers (by increasing number of vertices)
Category:4-dimensional regular polytope numbers
Category:Regular polychoron numbers
Regular polychoron numbers
4-dimensional regular polytope numbers (REDIRECT to Regular polychoron numbers)
 Pentachoron numbers (5-celled Regular polychoron numbers or 4-D Simplicial polytopic numbers)
 Tetracross/tesseract numbers (dual pair) Tetracross numbers (16-celled Regular polychoron numbers or 4-D Orthoplicial polytopic numbers) Tesseract numbers (8-celled Regular polychoron numbers or 4-D Regular orthotopic numbers)
 24-cell numbers (REDIRECT to 24-celled Regular polychoron numbers)
 Hypericosahedral/hyperdodecahedral numbers (dual pair) Hypericosahedral numbers or 600-cell numbers (both REDIRECT to 600-celled Regular polychoron numbers) Hyperdodecahedral numbers or 120-cell numbers (both REDIRECT to 120-celled Regular polychoron numbers)
Category:Centered regular polychoron numbers
Centered regular polychoron numbers
Centered 4-dimensional regular polytope numbers (REDIRECT to Centered regular polychoron numbers)
 Centered tetracross/tesseract numbers (dual pair) Centered tetracross numbers (16-celled Centered regular polychoron numbers or 4-D Centered orthoplicial polytopic numbers) Centered tesseract numbers (8-celled Centered regular polychoron numbers or 4-D Centered regular orthotopic numbers)
 Centered 24-cell numbers (REDIRECT to 24-celled Centered regular polychoron numbers)
 Centered hypericosahedral/hyperdodecahedral numbers (dual pair) Centered hypericosahedral numbers or Centered 600-cell numbers (both REDIRECT to 600-celled Centered regular polychoron numbers) Centered hyperdodecahedral numbers or Centered 120-cell numbers (both REDIRECT to 120-celled Centered regular polychoron numbers)