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Polygons which have an accompanying imaginary dimension for each real dimension
In geometry, a regular complex polygon is a generalization of a regular polygon in real space to an analogous structure in a complex Hilbert space, where
Regular_complex_polygon
Generalization of a polytope in real space
points lie at the vertices of a regular polygon centered on the origin. Three real projections of regular complex polygon 4{4}2 are illustrated above, with
Complex_polytope
Polygon in complex space, or which self-intersects
graphics, a polygon whose boundary is not simple. In geometry, a complex polygon is a polygon in the complex Hilbert plane, which has two complex dimensions
Complex_polygon
Polygonal chain whose vertices are not all coplanar
all coplanar, we speak of a plane polygon, otherwise a skew polygon." Regular complex polytopes, p. 6 Abstract Regular Polytopes, p.217 McMullen, Peter;
Skew_polygon
Plane figure bounded by line segments
Hilbert plane consisting of two complex dimensions. Star polygon: a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped
Polygon
Skew polygon derived from a polytope
the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive
Petrie_polygon
Poset representing certain properties of a polytope
group, not the (geometric) symmetry group. For example, any abstract polygon is regular, since angles, edge-lengths, edge curvature, skewness etc. do not
Abstract_polytope
Regular polygon that can be constructed with compass and straightedge
mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible
Constructible_polygon
Method of drawing geometric objects
same area as a given polygon, and regular polygons of 3, 4, or 5 sides (or one with twice the number of sides of a given polygon). But they could not
Straightedge and compass construction
Straightedge_and_compass_construction
In geometry, the Möbius–Kantor polygon is a regular complex polygon 3{3}3, , in C 2 {\displaystyle \mathbb {C} ^{2}} . 3{3}3 has 8 vertices, and 8 edges
Möbius–Kantor_polygon
Pictorial representation of symmetry
A regular complex polygon in C 2 {\displaystyle \mathbb {C} ^{2}} , has the form p{q}r or Coxeter diagram . The symmetry group of a regular complex polygon
Coxeter–Dynkin_diagram
Four-dimensional analogues of the regular polyhedra in three dimensions
and the regular polygons in two dimensions. There are six convex and ten star regular 4-polytopes, giving a total of sixteen. The convex regular 4-polytopes
Regular_4-polytope
Polygon shape with eight sides
Greek ὀκτάγωνον (oktágōnon) 'eight angles') is an eight-sided polygon or 8-gon. A regular octagon has Schläfli symbol {8} and can also be constructed as
Octagon
a regular polygon. The polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A p-gonal regular polygon is
List_of_regular_polytopes
Convex polyhedron with regular faces
a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons and that is not a uniform polyhedron. There are 92 such solids: 48
Johnson_solid
Polytope with highest degree of symmetry
themselves regular polytopes of dimension j≤ n. Regular polytopes are the generalised analog in any number of dimensions of regular polygons (for example
Regular_polytope
Polyhedron with regular congruent polygons as faces
A regular polyhedron is a polyhedron with regular and congruent polygons as faces. Its symmetry group acts transitively on its flags. A regular polyhedron
Regular_polyhedron
Non-planar polygon with infinitely many sides
M.; Regular complex polytopes (1974). Chapter 1. Regular polygons, 1.5. Regular polygons in n dimensions, 1.7. Zigzag and antiprismatic polygons, 1.8
Infinite_skew_polygon
Geometric object with flat sides
an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In
Polytope
Sub-list of the list of polytopes
icositetragram, with twenty four edges 257-gram, with two hundred and fifty seven edges List of regular polytopes and compounds § Stars Complex polygon
List of self-intersecting polygons
List_of_self-intersecting_polygons
Number of times a curve wraps around a point in the plane
the density is 1, by the Jordan curve theorem. By contrast, for a regular star polygon {p/q}, the density is q. Turning number cannot be defined for space
Winding_number
Points on a common circle
Every regular polygon is a cyclic polygon. For a cyclic polygon with an odd number of sides, all angles are equal if and only if the polygon is regular. A
Concyclic_points
unchanged when rotated 180 degrees. The idea is also applicable for regular complex polygons, p{q}r constructed in C 2 {\displaystyle \mathbb {C} ^{2}} : [
Configuration_(polytope)
connecting all pairs, just like a 5-simplex seen in projection. The regular complex polygon 2{4}3, also 3{ }+3{ } has 6 vertices in C 2 {\displaystyle \mathbb
3-3_duoprism
Polygon with equally angled vertices
it is a regular polygon. Isogonal polygons are equiangular polygons which alternate two edge lengths. For clarity, a planar equiangular polygon can be
Equiangular_polygon
Bipartite graph where each node of 1st set is linked to all nodes of 2nd set
Approach to Discrete Math, Springer, p. 437, ISBN 9780387941158. Coxeter, Regular Complex Polytopes, second edition, p.114 Garey, Michael R.; Johnson, David
Complete_bipartite_graph
Polygon associated with a compact Riemann surface
In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only information about
Fundamental_polygon
Shape with six sides
angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. A regular hexagon is defined as
Hexagon
Four-dimensional analog of the octahedron
be seen as its dual, a 4-4 duopyramid. The Möbius–Kantor polygon is a regular complex polygon 3{3}3, , in C 2 {\displaystyle \mathbb {C} ^{2}} shares the
16-cell
Extending the elements of a polytope to form a new figure
regular octahedron to obtain the stella octangula, a regular compound of two tetrahedra. Stellating a regular polygon symmetrically creates a regular
Stellation
Geometric configuration of 9 points and 12 lines
\3&6\\\end{smallmatrix}}\right]} , which is represented in the regular complex polygon 3{4}2, 9 vertices and 6 3-edges. It is also the dual configuration
Hesse_configuration
Developer of the Petrie polygon
known as the "Petrie polygon" and has many applications. The Petrie polygon of a regular polyhedron can be defined as the skew polygon (whose vertices do
John_Flinders_Petrie
Abstract regular 4-polytope with 4 cubic cells
edges added. It is also the complete bipartite graph K4,4, and the regular complex polygon 2{4}4, a generalized cross polytope.[clarification needed] The
Hemitesseract
Shape with four equal sides and angles
hyperbolic geometry both lack polygons with four equal sides and right angles, they have square-like regular polygons with four sides and other angles
Square
Regular object in four dimensional geometry
two from the northern hemisphere and two from the southern. The regular complex polygon 4{3}4, or contains the 24 vertices of the 24-cell, and 24 4-edges
24-cell
Decagram Hendecagram Dodecagram Icositetragram Concave polygon Cyclic polygon Regular polygon Polyform Gnomon Golygon List of uniform tilings Uniform
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Polygon with 65537 sides
a polygon with 65,537 (216 + 1) sides. The sum of the interior angles of any non-self-intersecting 65537-gon is 11,796,300°. The area of a regular 65537-gon
65537-gon
Uniform 6-polytope
forming the 222 honeycomb with this Coxeter-Dynkin diagram: . The regular complex polygon 3{3}3{3}3, , in C 2 {\displaystyle \mathbb {C} ^{2}} has a real
2_21_polytope
Graph operation
value of the complex number. For 3-regular graphs this norm is the T-number or triangulation number used in virology. The master polygon is an equilateral
Goldberg–Coxeter_construction
Natural number
the first regular polygon that does not tile the plane with copies of itself. The pentagon solid has the largest face of any of the five regular three-dimensional
5
Point where two or more curves, lines, or edges meet
point where two lines meet to form an angle and the point where edges of polygons and polyhedra meet are vertices. The vertex of an angle is the point where
Vertex_(geometry)
Four-dimensional analog of the icosahedron
4-polytope and honeycombs with icosahedron vertex figures: The regular complex polygons 3{5}3, and 5{3}5, , in C 2 {\displaystyle \mathbb {C} ^{2}} have
600-cell
Property of objects which are scaled or mirrored versions of each other
condition for similarity of polygons is that corresponding sides and diagonals are proportional. For given n, all regular n-gons are similar. Several
Similarity_(geometry)
Concept in geometry
circle as the limit of a sequence of regular polygons with an increasing number of sides. The area of a regular polygon is half its perimeter multiplied by
Area_of_a_circle
Pentagon with all sides equal but the angles may not be equal
In geometry, an equilateral pentagon is a polygon in the Euclidean plane with five sides of equal length. Its five vertex angles can take a range of sets
Equilateral_pentagon
Solid with four equal triangular faces
regular Platonic solids—polyhedra in which all of their faces are regular polygons. Known since antiquity, Platonic solids are named after the Greek philosopher
Regular_tetrahedron
Flat-sided three-dimensional shape
vertex is regular. A vertex-transitive polyhedron with regular polygonal faces is said to be uniform. This class includes the regular, quasi-regular, and semi-regular
Polyhedron
Natural number
twelve-sided polygon is a dodecagon. In its regular form, it is the largest polygon that can uniformly tile the plane alongside other regular polygons, as with
12_(number)
Self-intersecting uniform polyhedron
either star polygon faces, star polygon vertex figures, or both. The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called
Uniform_star_polyhedron
Overview of and topical guide to geometry
problem Parallel postulate Polygon Star polygon Pick's theorem Shape dissection Bolyai–Gerwien theorem Poncelet–Steiner theorem Polygon triangulation Pons asinorum
Outline_of_geometry
Number with a real and an imaginary part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary
Complex_number
Texture mapping technique
by generating a normal map from a high polygon model or height map. Normal maps are commonly stored as regular RGB images where the RGB components correspond
Normal_mapping
Four-dimensional analog of the dodecahedron
of every convex regular polytope in the first four dimensions (except the polygons {7} and above). As the sixth and largest regular convex 4-polytope
120-cell
Four-dimensional analogue of the cube
of regular 4-polytope and honeycombs, {4,3,p} with cubic cells. The regular complex polytope 4{4}2, , in C 2 {\displaystyle \mathbb {C} ^{2}} has a real
Tesseract
Natural number
number system. A polygon with eight sides is an octagon. A regular octagon can fill a plane-vertex with a regular triangle and a regular icositetragon,
8
Simple curve of Euclidean geometry
side of the polygon. Every regular polygon and every triangle is a tangential polygon. A cyclic polygon is any convex polygon about which a circle can be
Circle
Type of plane partition
Voronoi polygons, domain(s) of influence, Voronoi decomposition, Voronoi tessellation(s), Dirichlet tessellation(s). Voronoi tessellations of regular lattices
Voronoi_diagram
Concept in differential equation mathematics
In mathematics, in the theory of ordinary differential equations in the complex plane C {\displaystyle \mathbb {C} } , the points of C {\displaystyle \mathbb
Regular_singular_point
Construction on any polygon that yields a regular polygon with the same number of sides
arbitrary planar polygons. The theorem asserts that a certain procedure when applied to an arbitrary polygon always yields a regular polygon having the same
Petr–Douglas–Neumann_theorem
Form of an object
include: Circle, Square, Triangle, Rectangle, Oval, Star (polygon), Rhombus, Semicircle. Regular polygons starting at pentagon follow the naming convention of
Shape
Circle associated with a quadratic equation
circles have been used to develop ruler-and-compass constructions of regular polygons. Given a quadratic equation in the form x2 − sx + p = 0 the circle
Carlyle_circle
the empty Schläfli symbol {}. Polygon Equilateral Cyclic polygon Convex polygon Star polygon Pentagram Regular polygon Equilateral triangle Simplex Square
List_of_mathematical_shapes
Set of basic shapes which assemble into a polygon
of a polygon is a set of primitive units (e.g., triangles, rectangles, etc.), which do not overlap and whose union equals the polygon. A polygon partition
Polygon_partition
Polyhedron formed by joining mirroring pyramids base-to-base
regular polygon, the bipyramid is also called regular. A bipyramid is a polyhedron constructed by fusing two pyramids which share the same polygonal base;
Bipyramid
polychoron Convex regular polychoron Duocylinder Tesseract Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974). Regular Polytopes
3-4_duoprism
Study of complex manifolds and several complex variables
complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry
Complex_geometry
Solid with twenty equal triangular faces
The regular icosahedron (or simply icosahedron) is a convex polyhedron that can be constructed from a pentagonal antiprism by attaching two pentagonal
Regular_icosahedron
Feature of a polyhedron, polytope, etc.
authors call a facet of a polyhedron any polygon whose corners are vertices of the polyhedron, including polygons that are not faces. To facet a polyhedron
Facet_(geometry)
Way to divide polygon into smaller parts
dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric
Finite_subdivision_rule
Five tiles used in Islamic decorative art
An interlocking decagram-polygon mosaic design An interlocking decagram-polygon mosaic design An interlocking decagram-polygon mosaic design First, divide
Girih_tile
Property of a mathematical space
Graph (combinatorics) Real number Length 2 dimensions Plane Surface Polygon Net Complex number Cartesian coordinate system List of uniform tilings Area 3
Dimension
Solid with 12 equal pentagonal faces
of the Platonic solids, a set of polyhedrons in which the faces are regular polygons that are congruent and the same number of faces meet at a vertex. This
Regular_dodecahedron
Group of symmetries of a regular polygon
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among
Dihedral_group
Regular polytope dual to the hypercube in any number of dimensions
Petrie polygon projections map the points into a regular 2n-gon or lower order regular polygons. A second projection takes the 2(n−1)-gon petrie polygon of
Cross-polytope
In 4-dimensional complex geometry, the Witting polytope is a regular complex polytope, named as: 3{3}3{3}3{3}3, and Coxeter diagram . It has 240 vertices
Witting_polytope
Geometric objects with a common centre
well-defined centers can be concentric, including circles, spheres, regular polygons, regular polyhedra, parallelograms, cones, conic sections, and quadrics
Concentric_objects
Geometric structure of 8 points and 8 lines
the polygon vertices are complex numbers. Kantor's solution for p = 4 {\displaystyle p=4} , a pair of mutually-inscribed quadrilaterals in the complex projective
Möbius–Kantor_configuration
Set of polygons in mathematics
In mathematics Nef polygons and Nef polyhedra are the sets of polygons and polyhedra which can be obtained from a finite set of halfplanes (halfspaces)
Nef_polygon
In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32
Small complex icosidodecahedron
Small_complex_icosidodecahedron
Partition of a polygon into triangles of equal area
question is: Which polygons can be equidissected into how many pieces? Particular attention has been given to trapezoids, kites, regular polygons, centrally symmetric
Equidissection
Region between two concentric circles
A={\frac {\theta }{2}}\left(R^{2}-r^{2}\right).} In complex analysis an annulus ann(a; r, R) in the complex plane is an open region defined as r < | z − a
Annulus_(mathematics)
Set of principles for modeling solid geometry
customization) Creating polygon mesh models for rapid prototyping (to aid surgeons preparing for difficult surgeries, for example) Combining polygon mesh models with
Solid_modeling
quasirational polygon has all orbits bounded. The notion of quasirational is technical (see references) but it includes the class of regular polygons and convex
Outer_billiards
symmetry as a regular complex polyhedron with 72 vertices, 216 3{} edges, 54 3{3}3 faces. Its vertex figure is 3{4}2, and van oss polygon 3{4}3. It is
Hessian_polyhedron
Convex polytope, the n-dimensional analogue of a square and a cube
vertex figure are regular simplexes. The regular polygon perimeter seen in these orthogonal projections is called a Petrie polygon. The generalized squares
Hypercube
Tiling of a plane by regular hexagons and equilateral triangles
uniform tilings of the Euclidean plane by regular polygons. It consists of equilateral triangles and regular hexagons, arranged so that each hexagon is
Trihexagonal_tiling
Type of spatial anti-aliasing
calculates interior polygon fragments only once per pixel, aliasing and other artifacts will still be visible inside rendered polygons where fragment shader
Multisample_anti-aliasing
Four-dimensional geometric object with flat sides
polygons and Kepler–Poinsot polyhedra. A 4-polytope is regular if it is transitive on its flags. This means that its cells are all congruent regular polyhedra
4-polytope
Smallest convex set containing a given set
convex hull of a simple polygon encloses the given polygon and is partitioned by it into regions, one of which is the polygon itself. The other regions
Convex_hull
Conformal mappings in complex analysis
Schwarz–Christoffel transformation. Through the theory of complex ordinary differential equations with regular singular points and the Schwarzian derivative, the
Schwarz_triangle_function
Polyhedron with non-planar faces
called regular skew apeirohedra. According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to
Regular_skew_polyhedron
Power series with rational exponents
ramified case, where m > 1, and the regular case where m = 1. The way of applying recursively the method of the Newton polygon has been described precedingly
Puiseux_series
Mathematical instrument consisting of two hinged rulers
a "polymetric compass" c. 1670, including a scale for constructing regular polygons. The Italian astronomer Galileo Galilei added further scales in the
Sector_(instrument)
7-dimensional hypercube
(iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5), ISBN 0-486-61480-8 Coxeter, H.S.M. (1991) [1974]. Regular Complex Polytopes
7-cube
Regular tiling of a two-dimensional space
lattice Hexagonal prismatic honeycomb Tilings of regular polygons List of uniform tilings List of regular polytopes Hexagonal tiling honeycomb Hex map board
Hexagonal_tiling
3D shape made of polyhedra sharing a common center
polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be
Polytope_compound
Group homomorphism into the general linear group over a vector space
which the vector space is defined. The most important case is the field of complex numbers. The other important cases are the field of real numbers, finite
Group_representation
Symmetric bipartite cubic graph with 16 vertices and 24 edges
edges belong to the complex projective plane. That is, in Kantor's solution, the coordinates of the polygon vertices are complex numbers. Kantor's solution
Möbius–Kantor_graph
Entertainment venue in Brussels, Belgium
building has a circular appearance, but in fact is constructed as a regular polygon. It can hold 2,000 spectators, and nowadays is primarily used for live
Cirque_Royal
Branch of geometry that studies combinatorial properties and constructive methods
basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these
Discrete_geometry
REGULAR COMPLEX-POLYGON
REGULAR COMPLEX-POLYGON
Girl/Female
Muslim
One who remembers Allah regularly
Surname or Lastname
English (Yorkshire)
English (Yorkshire) : habitational name from any of various places called Copley, for example in County Durham, Staffordshire, and Yorkshire, from the Old English personal name Coppa (apparently a byname for a tall man) or from copp ‘hilltop’ + lēah ‘woodland clearing’.
Boy/Male
Hindu, Indian, Traditional
Conduct; Regular Performance of Worship
Girl/Female
Arabic, Muslim
Complex; Zigzag; Curling
Surname or Lastname
English
English : habitational name from Coppull in Lancashire, recorded in the 13th century as Cophill, from Old English copp ‘peak’ + hyll ‘hill’.English : nickname from Old French curt peil ‘short hair’.Probably an Americanized spelling of German and Jewish Koppel or German and Dutch Kappel.
Girl/Female
Hebrew
Precious.
Girl/Female
Hindu, Indian
Complex
Girl/Female
Bengali, Indian
Good Complex
Girl/Female
Arabic, Muslim
Pilgrimage to Makkah Other than Regular Hajj Days
Surname or Lastname
English
English : unexplained.Americanized form of German Koppler.
Boy/Male
Hindu, Indian, Tamil
Regular Winner
Girl/Female
Muslim
Complex, Zigzag, Curling
Boy/Male
Tamil
Poornan | பூரà¯à®¨à®¾à®¨
Complete
Poornan | பூரà¯à®¨à®¾à®¨
Girl/Female
Indian
One who remembers Allah regularly
Surname or Lastname
English
English : habitational name, probably from Comley in Shropshire or Combley on the Isle of Wight; both are named with Old English cumb ‘valley’ + lēah ‘woodland clearing’.
Boy/Male
Tamil
Complete
Boy/Male
Indian, Sanskrit
Connector; Regulator
Girl/Female
Muslim/Islamic
One who remembers Allah regularly
Girl/Female
Tamil
Complete
Boy/Male
Gujarati, Haryanvi, Hindu, Indian, Kannada, Marathi, Telugu
Regular; Ethical; Good in Nature
REGULAR COMPLEX-POLYGON
REGULAR COMPLEX-POLYGON
Girl/Female
English
Modernand Jennifer.
Boy/Male
Hindu, Indian, Punjabi, Sikh
Fragrance Like Sandalwood; Full of Fragrance
Boy/Male
Hindu, Indian
Written; Beautiful Writing
Boy/Male
French
Friend.
Boy/Male
Tamil
The Moon
Girl/Female
Indian, Telugu
Well Known; Origin
Girl/Female
English American
Young deer. The Greek mythological deity of fertility and nature was Fauna. She was famous for...
Boy/Male
Arabic
Of Allah.
Girl/Female
Muslim/Islamic
She was a narrator of hadith
Surname or Lastname
English, Scottish, and Irish (of Norman origin)
English, Scottish, and Irish (of Norman origin) : variant of Cumming.
REGULAR COMPLEX-POLYGON
REGULAR COMPLEX-POLYGON
REGULAR COMPLEX-POLYGON
REGULAR COMPLEX-POLYGON
REGULAR COMPLEX-POLYGON
adv.
In a complex manner; not simply.
a.
Repeatedly compound; made up of complex constituents.
a.
Having all the parts of the same kind alike in size and shape; as, a regular flower; a regular sea urchin.
a.
Not regular; not bound by monastic vows or rules; not confined to a monastery, or subject to the rules of a religious community; as, a secular priest.
pl.
of Tegula
n.
A complex; an aggregate of parts; a complication.
a.
Governed by rule or rules; steady or uniform in course, practice, or occurence; not subject to unexplained or irrational variation; returning at stated intervals; steadily pursued; orderlly; methodical; as, the regular succession of day and night; regular habits.
n.
Composed of two or more parts; composite; not simple; as, a complex being; a complex idea.
a.
Conformed to a rule; agreeable to an established rule, law, principle, or type, or to established customary forms; normal; symmetrical; as, a regular verse in poetry; a regular piece of music; a regular verb; regular practice of law or medicine; a regular building.
a.
Intricate; entangled; complicated; complex.
a.
Complex, complicated.
a.
Constituted, selected, or conducted in conformity with established usages, rules, or discipline; duly authorized; permanently organized; as, a regular meeting; a regular physican; a regular nomination; regular troops.
a.
Thorough; complete; unmitigated; as, a regular humbug.
n.
One who is not regular; especially, a soldier not in regular service.
imp. & p. p.
of Compile
imp. & p. p.
of Couple
a.
Belonging to a monastic order or community; as, regular clergy, in distinction dfrom the secular clergy.
a.
Not complex; uncompounded; simple.
a.
Not regular; not conforming to a law, method, or usage recognized as the general rule; not according to common form; not conformable to nature, to the rules of moral rectitude, or to established principles; not normal; unnatural; immethodical; unsymmetrical; erratic; no straight; not uniform; as, an irregular line; an irregular figure; an irregular verse; an irregular physician; an irregular proceeding; irregular motion; irregular conduct, etc. Cf. Regular.
pl.
of Regulus