Egg
The egg is the organic vessel containing the zygote in which an embryo develops until it can survive on its own. An egg results from fertilization of an egg cell. Most arthropods and mollusks lay eggs, although some, such as scorpions do not. Reptile eggs, bird eggs, monotreme eggs are laid out of water, are surrounded by a protective shell, either flexible or inflexible. Eggs laid on land or in nests are kept within a warm and favorable temperature range while the embryo grows; when the embryo is adequately developed it hatches, i.e. breaks out of the egg's shell. Some embryos have a temporary egg tooth they use to pip, or break the eggshell or covering; the largest recorded egg is from a whale shark, was 30 cm × 14 cm × 9 cm in size. Whale shark eggs hatch within the mother. At 1.5 kg and up to 17.8 cm × 14 cm, the ostrich egg is the largest egg of any living bird, though the extinct elephant bird and some dinosaurs laid larger eggs. The bee hummingbird produces the smallest known bird egg; some eggs laid by reptiles and most fish, amphibians and other invertebrates can be smaller.
Reproductive structures similar to the egg in other kingdoms are termed "spores," or in spermatophytes "seeds," or in gametophytes "egg cells". Several major groups of animals have distinguishable eggs; the most common reproductive strategy for fish is known as oviparity, in which the female lays undeveloped eggs that are externally fertilized by a male. Large numbers of eggs are laid at one time and the eggs are left to develop without parental care; when the larvae hatch from the egg, they carry the remains of the yolk in a yolk sac which continues to nourish the larvae for a few days as they learn how to swim. Once the yolk is consumed, there is a critical point after which they must learn how to hunt and feed or they will die. A few fish, notably the rays and most sharks use ovoviviparity in which the eggs are fertilized and develop internally; however the larvae still grow inside the egg consuming the egg's yolk and without any direct nourishment from the mother. The mother gives birth to mature young.
In certain instances, the physically most developed offspring will devour its smaller siblings for further nutrition while still within the mother's body. This is known as intrauterine cannibalism. In certain scenarios, some fish such as the hammerhead shark and reef shark are viviparous, with the egg being fertilized and developed internally, but with the mother providing direct nourishment; the eggs of fish and amphibians are jellylike. Cartilagenous fish eggs are fertilized internally and exhibit a wide variety of both internal and external embryonic development. Most fish species spawn eggs that are fertilized externally with the male inseminating the eggs after the female lays them; these eggs would dry out in the air. Air-breathing amphibians lay their eggs in water, or in protective foam as with the Coast foam-nest treefrog, Chiromantis xerampelina. Bird eggs are incubated for a time that varies according to the species. Average clutch sizes range from one to about 17; some birds lay eggs when not fertilized.
The default color of vertebrate eggs is the white of the calcium carbonate from which the shells are made, but some birds passerines, produce colored eggs. The pigment biliverdin and its zinc chelate give a green or blue ground color, protoporphyrin produces reds and browns as a ground color or as spotting. Non-passerines have white eggs, except in some ground-nesting groups such as the Charadriiformes and nightjars, where camouflage is necessary, some parasitic cuckoos which have to match the passerine host's egg. Most passerines, in contrast, lay colored eggs if there is no need of cryptic colors; however some have suggested that the protoporphyrin markings on passerine eggs act to reduce brittleness by acting as a solid state lubricant. If there is insufficient calcium available in the local soil, the egg shell may be thin in a circle around the broad end. Protoporphyrin speckling compensates for this, increases inversely to the amount of calcium in the soil. For the same reason eggs in a clutch are more spotted than early ones as the female's store of calcium is depleted.
The color of individual eggs is genetically influenced, appears to be inherited through the mother only, suggesting that the gene responsible for pigmentation is on the sex determining W chromosome. It used to be thought that color was applied to the shell before laying, but this research shows that coloration is an integral part of the development of the shell, with the same protein responsible for depositing calcium carbonate, or protoporphyrins when there is a lack of that mineral. In species such as the common guillemot, which nest in large groups, each female's eggs have different markings, making it easier for females to identify their own eggs on the crowded cliff ledges on which they breed. Bird eggshells are diverse. For example: cormorant eggs are rough and chalky tinamou eggs are shiny duck eggs are oily and waterproof cassowary eggs are pittedTiny pores in bird eggshells allow the embryo to breathe; the domestic
Cassini oval
A Cassini oval is a quartic plane curve defined as the set of points in the plane such that the product of the distances to two fixed points is constant. This may be contrasted with an ellipse, for which the sum of the distances is constant, rather than the product. Cassini ovals are the special case of polynomial lemniscates when the polynomial used has degree 2. Cassini ovals are named after the astronomer Giovanni Domenico Cassini who studied them in 1680. Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one focus of the oval. Other names include Cassinian curves and ovals of Cassini. A Cassini oval is a set of points, such that for any point P of the set, the product of the distances | P P 1 |, | P P 2 | to two fixed points P 1, P 2, is constant denoted by b 2, b > 0,:. As with an ellipse, the fixed points P 1, P 2 are called the foci of the Cassini oval. If the foci are and the equation of the curve is = b 4; when expanded this becomes 2 − 2 a 2 + a 4 = b 4.
The equivalent polar equation is r 4 − 2 a 2 r 2 cos 2 θ = b 4 − a 4. The curve depends, up to similarity, on e = b/a; when e < 1, the curve consists of two disconnected loops, each of which contains a focus. When e = 1, the curve is the lemniscate of Bernoulli having the shape of a sideways figure eight with a double point at the origin; when e > 1, the curve is a connected loop enclosing both foci. It is peanut-shaped for 1 < e < 2 and convex for e ≥ 2. The limiting case of a → 0, in which case the foci coincide with each other, is a circle; the curve always has x-intercepts at ±c where c2 = a2 + b2. When e < 1 there are two additional real x-intercepts and when e > 1 there are two real y-intercepts, all other x and y-intercepts being imaginary. The curve has double points at the circular points at infinity, in other words the curve is bicircular; these points are biflecnodes, meaning that the curve has two distinct tangents at these points and each branch of the curve has a point of inflection there.
From this information and Plücker's formulas it is possible to deduce the Plücker numbers for the case e ≠ 1: degree = 4, class = 8, number of nodes = 2, number of cusps = 0, number of double tangents = 8, number of points of inflection = 12, genus = 1. The tangents at the circular points are given by x ± iy = ±a which have real points of intersection at. So the foci are, in fact, foci in the sense defined by Plücker; the circular points are points of inflection. When e ≠ 1 the curve has class eight, which implies that there should be at total of eight real foci. Six of these have been accounted for in the two triple foci and the remaining two are at. So the additional foci are on the x-axis when the curve has two loops and on the y-axis when the curve has a single loop. Orthogonal trajectories of a given pencil of curves are curves which intersect all given curves orthogonally. For example the orthogonal trajectories of a pencil of confocal ellipses are the confocal hyperbolas with the same foci.
For Cassini ovals one has: The orthogonal trajectories of the Cassini curves with foci P 1, P 2 are the equilateral hyperbolas containing P 1, P 2 with the same center as the Cassini ovals. Proof: For simplicity one chooses P
Ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more of an affine transformation. An ellipsoid is a quadric surface. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties; every planar cross section is empty, or is reduced to a single point. It is bounded. An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid; the line segments that are delimited on the axes of symmetry by the ellipsoid are called the principal axes, or axes of the ellipsoid. If the three axes have different lengths, the ellipsoid is said to be tri-axial or scalene, the axes are uniquely defined. If two of the axes have the same length the ellipsoid is an "ellipsoid of revolution" called a spheroid. In this case, the ellipsoid is invariant under a rotation around the third axis, there are thus infinitely many ways of choosing the two perpendicular axes of the same length.
If the third axis is shorter, the ellipsoid is an oblate spheroid. If the three axes have the same length, the ellipsoid is a sphere. Using a Cartesian coordinate system in which the origin is the center of the ellipsoid and the coordinate axes are axes of the ellipsoid, the implicit equation of the ellipsoid has the standard form x 2 a 2 + y 2 b 2 + z 2 c 2 = 1, where a, b, c are positive real numbers; the points, lie on the surface. The line segments from the origin to these points are called the principal semi-axes of the ellipsoid, because a, b, c are half the length of the principal axes, they correspond to semi-minor axis of an ellipse. If a = b > c, one has an oblate spheroid. The ellipsoid may be parameterized in several ways, which are simpler to express when the ellipsoid axes coincide with coordinate axes. A common choice is x = a cos cos , y = b cos sin , z = c sin , where − π 2 ≤ θ ≤ π 2, − π ≤ φ ≤ π; these parameters may be interpreted as spherical coordinates, where π / 2 − θ is the polar angle, φ is the azimuth angle of the point of the ellipsoid.
The volume bounded by the ellipsoid is V = 4 3 π a b c. Alternatively expressed, where A, B and C are the lengths of the principal semi-axes: V = π 6 A B C ≈ 0.523 A B C. Note that this equation reduces to that of the volume of a sphere when all three elliptic radii are equal, to that of an oblate or prolate spheroid when two of them are equal; the volume of an ellipsoid is 2 3 the volume of a circumscribed elliptic cylinder, π 6 the volume of the circumscribed box. The volumes of the inscribed and circumscribed boxes are respectively: V inscribed = 8 3 3 a b c, V circumscribed = 8 a b c; the surface area of a general ellipsoid is S = 2 π c 2 + 2 π a b sin ( E ( φ
Oval (projective plane)
In projective geometry an oval is a circle-like pointset in a plane, defined by incidence properties. The standard examples are the nondegenerate conics. However, a conic is only defined in a pappian plane, whereas an oval may exist in any type of projective plane. In the literature, there are many criteria which imply that an oval is a conic, but there are many examples, both infinite and finite, of ovals in pappian planes which are not conics; as mentioned, in projective geometry an oval is defined by incidence properties, but in other areas, ovals may be defined to satisfy other criteria, for instance, in differential geometry by differentiability conditions in the real plane. The higher dimensional analog of an oval is an ovoid in a projective space. A generalization of the oval concept is an abstract oval, a structure, not embedded in a projective plane. Indeed, there exist abstract ovals. In a projective plane a set Ω of points is called an oval, if:Any line l meets Ω in at most two points, For any point P ∈ Ω there exists one tangent line t through P, i.e. t ∩ Ω =.
When |l ∩ Ω| = 0 the line l is an exterior line, if |l ∩ Ω| = 1 a tangent line and if |l ∩ Ω| = 2 the line is a secant line. For finite planes we have a more convenient characterization: For a finite projective plane of order n a set Ω of points is an oval if and only if |Ω| = n + 1 and no three points are collinear. A set of points in an affine plane satisfying the above definition is called an affine oval. An affine oval is always a projective oval in the projective closure of the underlying affine plane. An oval can be considered as a special quadratic set. In any pappian projective plane there exist nondegenerate projective conic sections and any nondegenerate projective conic section is an oval; this statement can be verified by a straightforward calculation for any of the conics. Non degenerate conics are ovals with special properties: Pascal's Theorem and its various degenerations are valid. There are many projectivities. In the real planeIf one glues one half of a circle and a half of an ellipse smoothly together, one gets a non-conic oval.
If one takes the inhomogeneous representation of a conic oval as a parabola plus a point at infinity and replaces the expression x2 by x4, one gets an oval, not a conic. If one takes the inhomogeneous representation of a conic oval as a hyperbola plus two points at infinity and replaces the expression 1/x by 1/x3, one gets an oval, not a conic; the implicit curve x4 + y4 = 1 is a non conic oval.in a finite plane of orderIn a finite pappian plane of order a nondegenerate conic has a nucleus, which can be exchanged with any point of the conic to obtain an oval, not a conic. For the field K = GF with 2m elements let Ω = ∪ For k ∈ and k and m coprime, the set Ω is an oval, not a conic. Further finite examples can be found here: For an oval to be a conic the oval and/or the plane has to fulfill additional conditions. Here are some results: An oval in an arbitrary projective plane, which fulfills the incidence condition of Pascal's theorem or the 5-point degeneration of it, is a nondegenerate conic.
If Ω is an oval in a pappian projective plane and the group of projectivities which leave Ω invariant is 3-transitive, i.e. for 2 triples A1, A2, A3. In the finite case 2-transitive is sufficient. An oval Ω in a pappian projective plane of characteristic ≠ 2 is a conic if and only if for any point P of a tangent there is an involutory perspectivity with center P which leaves Ω invariant. If Ω is an oval in a finite Desarguesian projective plane of odd order, PG Ω is a conic; this implies that, after a possible change of coordinates, every oval of PG with q odd has the parametrization: ∪. For topological ovals the following simple criteria holds: 5. Any closed oval of the complex projective plane is a conic. An oval in a finite projective plane of order q is a -arc, in other words, a set of q + 1 points, no three collinear. Ovals in the Desarguesian projective plane PG for q odd are just the nonsingular conics. However, ovals in PG for q have not yet been classified. In an arbitrary finite projective plane of odd order q, no sets with more points than q + 1, no three of which are collinear, exist, as first pointed out by Bose in a 1947 paper on applications of this sort of mathematics to the statistical design of experiments.
Furthermore, by Qvist's theorem, through any point not on an oval there pass either zero or two tangent lines of that oval. When q is the situation is different. In this case, sets of q + 2 points, no three of which collinear, may exist in a finite project
Reflection symmetry
Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure, indistinguishable from its transformed image is called mirror symmetric. In formal terms, a mathematical object is symmetric with respect to a given operation such as reflection, rotation or translation, if, when applied to the object, this operation preserves some property of the object; the set of operations that preserve a given property of the object form a group. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations; the symmetric function of a two-dimensional figure is a line such that, for each perpendicular constructed, if the perpendicular intersects the figure at a distance'd' from the axis along the perpendicular there exists another intersection of the shape and the perpendicular, at the same distance'd' from the axis, in the opposite direction along the perpendicular.
Another way to think about the symmetric function is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other's mirror images. Thus a square has four axes of symmetry, because there are four different ways to fold it and have the edges all match. A circle has infinitely many axes of symmetry. Triangles with reflection symmetry are isosceles. Quadrilaterals with reflection symmetry are kites, deltoids and isosceles trapezoids. All even-sided polygons have two simple reflective forms, one with lines of reflections through vertices, one through edges. For an arbitrary shape, the axiality of the shape measures how close it is to being bilaterally symmetric, it equals 1 for shapes with reflection symmetry, between 2/3 and 1 for any convex shape. For each line or plane of reflection, the symmetry group is isomorphic with Cs, one of the three types of order two, hence algebraically C2; the fundamental domain is a half-space. In certain contexts there is rotational as well as reflection symmetry.
Mirror-image symmetry is equivalent to inversion symmetry. For more general types of reflection there are correspondingly more general types of reflection symmetry. For example: with respect to a non-isometric affine involution with respect to circle inversion. Animals that are bilaterally symmetric have reflection symmetry in the sagittal plane, which divides the body vertically into left and right halves, with one of each sense organ and limb pair on either side. Most animals are bilaterally symmetric because this supports forward movement and streamlining. Mirror symmetry is used in architecture, as in the facade of Santa Maria Novella, Venice, it is found in the design of ancient structures such as Stonehenge. Symmetry was a core element in some styles such as Palladianism. Patterns in nature Point reflection symmetry Stewart, Ian. What Shape is a Snowflake? Magical Numbers in Nature. Weidenfeld & Nicolson. Weyl, Hermann. Symmetry. Princeton: Princeton University Press. ISBN 0-691-02374-3. Mapping with symmetry - source in Delphi Reflection Symmetry Examples from Math Is Fun
