Quadrant Model of Reality: Quadrant Model of Reality Book 10 Philosophy

Philosophy Chapter

QMR Quadrants are perpendicular. In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects.

A line is said to be perpendicular to another line if the two lines intersect at a right angle.[1] Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order.

Perpendicularity easily extends to segments and rays. For example, a line segment \overline{AB} is perpendicular to a line segment \overline{CD} if, when each is extended in both directions to form an infinite line, these two resulting lines are perpendicular in the sense above. In symbols, \overline{AB} \perp \overline{CD} means line segment AB is perpendicular to line segment CD.[2] The point B is called a foot of the perpendicular from A to segment \overline{CD}, or simply, a foot of A on \overline{CD}.[3]

A line is said to be perpendicular to a plane if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines.

Two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle (90 degrees).

Perpendicularity is one particular instance of the more general mathematical concept of orthogonality; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its normal.

QMRDetermining congruence

Sufficient evidence for congruence between two triangles in Euclidean space can be shown through the following four comparisons:

SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

The ASA Postulate was contributed by Thales of Miletus (Greek). In most systems of axioms, the three criteria—SAS, SSS and ASA—are established as theorems. In the School Mathematics Study Group system SAS is taken as one (#15) of 22 postulates.

AAS (Angle-Angle-Side): If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. (In British usage, ASA and AAS are usually combined into a single condition AAcorrS - any two angles and a corresponding side.)[3] For American usage, AAS is equivalent to an ASA condition, by the fact that if any two angles are given, so is the third angle, since their sum should be 180°.

RHS (Right-angle-Hypotenuse-Side), also known as HL (Hypotenuse-Leg): If two right-angled triangles have their hypotenuses equal in length, and a pair of shorter sides are equal in length, then the triangles are congruent.

For two polygons to be congruent, they must have an equal number of sides (and hence an equal number—the same number—of vertices). Two polygons with n sides are congruent if and only if they each have numerically identical sequences (even if clockwise for one polygon and counterclockwise for the other) side-angle-side-angle-... for n sides and n angles.

Congruence of polygons can be established graphically as follows:

First, match and label the corresponding vertices of the two figures.

Second, draw a vector from one of the vertices of the one of the figures to the corresponding vertex of the other figure. Translate the first figure by this vector so that these two vertices match.

Third, rotate the translated figure about the matched vertex until one pair of corresponding sides matches.

Fourth, reflect the rotated figure about this matched side until the figures match.

If at any time the step cannot be completed, the polygons are not congruent.

Side-side-angle

The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not by itself prove congruence. In order to show congruence, additional information is required such as the measure of the corresponding angles and in some cases the lengths of the two pairs of corresponding sides. There are a few possible cases:

If two triangles satisfy the SSA condition and the length of the side opposite the angle is greater than or equal to the length of the adjacent side (SsA, or long side-short side-angle), then the two triangles are congruent. The opposite side is sometimes longer when the corresponding angles are acute, but it is always longer when the corresponding angles are right or obtuse. Where the angle is a right angle, also known as the Hypotenuse-Leg (HL) postulate or the Right-angle-Hypotenuse-Side (RHS) condition, the third side can be calculated using the Pythagorean Theorem thus allowing the SSS postulate to be applied.

If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is equal to the length of the adjacent side multiplied by the sine of the angle, then the two triangles are congruent.

If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is greater than the length of the adjacent side multiplied by the sine of the angle (but less than the length of the adjacent side), then the two triangles cannot be shown to be congruent. This is the ambiguous case and two different triangles can be formed from the given information, but further information distinguishing them can lead to a proof of congruence.

Angle-angle-angle

In Euclidean geometry, AAA (Angle-Angle-Angle) (or just AA, since in Euclidean geometry the angles of a triangle add up to 180°) does not provide information regarding the size of the two triangles and hence proves only similarity and not congruence in Euclidean space.

However, in spherical geometry and hyperbolic geometry (where the sum of the angles of a triangle varies with size) AAA is sufficient for congruence on a given curvature of surface.[4]

QMRIn mathematics, a quadratrix (from the Latin word quadrator, squarer) is a curve having ordinates which are a measure of the area (or quadrature) of another curve. The two most famous curves of this class are those of Dinostratus and E. W. Tschirnhausen, which are both related to the circle.

QMRIn relationship to parallel lines[edit]

The arrowhead marks indicate that the lines a and b, cut by the transversal line c, are parallel.

If two lines (a and b) are both perpendicular to a third line (c), all of the angles formed along the third line are right angles. Therefore, in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.

In the figure at the right, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because vertical angles are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent. Therefore, if lines a and b are parallel, any of the following conclusions leads to all of the others:

One of the angles in the diagram is a right angle.

One of the orange-shaded angles is congruent to one of the green-shaded angles.

Line c is perpendicular to line a.

Line c is perpendicular to line b.

In computing distances[edit]

The distance from a point to a line is the distance to the nearest point on that line. That is the point at which a segment from it to the given point is perpendicular to the line.

Likewise, the distance from a point to a curve is measured by a line segment that is perpendicular to a tangent line to the curve at the nearest point on the curve.

Perpendicular regression fits a line to data points by minimizing the sum of squared perpendicular distances from the data points to the line.

The distance from a point to a plane is measured as the length from the point along a segment that is perpendicular to the plane, meaning that it is perpendicular to all lines in the plane that pass through the nearest point in the plane to the given point.

Graph of functions[edit]

In the two-dimensional plane, right angles can be formed by two intersected lines which the product of their slopes equals −1. Thus defining two linear functions: y1 = a1x + b1 and y2 = a2x + b2, the graphs of the functions will be perpendicular and will make four right angles where the lines intersect if and only if a1a2 = −1. However, this method cannot be used if the slope is zero or undefined (the line is parallel to an axis).

For another method, let the two linear functions: a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0. The lines will be perpendicular if and only if a1a2 + b1b2 = 0. This method is simplified from the dot product (or, more generally, the inner product) of vectors. In particular, two vectors are considered orthogonal if their inner product is zero.

In circles and other conics[edit]

Circles[edit]

Each diameter of a circle is perpendicular to the tangent line to that circle at the point where the diameter intersects the circle.

A line segment through a circle's center bisecting a chord is perpendicular to the chord.

If the intersection of any two perpendicular chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then a2 + b2 + c2 + d 2 equals the square of the diameter.[4]

The sum of the squared lengths of any two perpendicular chords intersecting at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8r 2 – 4p 2 (where r is the circle's radius and p is the distance from the center point to the point of intersection).[5]

Thales' theorem states that two lines both through the same point on a circle but going through opposite endpoints of a diameter are perpendicular.

Ellipses[edit]

The major and minor axes of an ellipse are perpendicular to each other and to the tangent lines to the ellipse at the points where the axes intersect the ellipse.

The major axis of an ellipse is perpendicular to the directrix and to each latus rectum.

Parabolas[edit]

In a parabola, the axis of symmetry is perpendicular to each of the latus rectum, the directrix, and the tangent line at the point where the axis intersects the parabola.

From a point on the tangent line to a parabola's vertex, the other tangent line to the parabola is perpendicular to the line from that point through the parabola's focus.

The orthoptic property of a parabola is that If two tangents to the parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect on the directrix are perpendicular.

Hyperbolas[edit]

The transverse axis of a hyperbola is perpendicular to the conjugate axis and to each directrix.

The product of the perpendicular distances from a point P on a hyperbola or on its conjugate hyperbola to the asymptotes is a constant independent of the location of P.

A rectangular hyperbola has asymptotes that are perpendicular to each other. It has an eccentricity equal to \sqrt{2}.

In polygons[edit]

Triangles[edit]

The legs of a right triangle are perpendicular to each other.

The altitudes of a triangle are perpendicular to their respective bases. The perpendicular bisectors of the sides also play a prominent role in triangle geometry.

The Euler line of an isosceles triangle is perpendicular to the triangle's base.

The Droz-Farny line theorem concerns a property of two perpendicular lines intersecting at a triangle's orthocenter.

Harcourt's theorem concerns the relationship of line segments through a vertex and perpendicular to any line tangent to the triangle's incircle.

Quadrilaterals[edit]

In a square or other rectangle, all pairs of adjacent sides are perpendicular. A right trapezoid is a trapezoid that has two pairs of adjacent sides that are perpendicular.

Each of the four maltitudes of a quadrilateral is a perpendicular to a side through the midpoint of the opposite side.

An orthodiagonal quadrilateral is a quadrilateral whose diagonals are perpendicular. These include the square, the rhombus, and the kite. By Brahmagupta's theorem, in an orthodiagonal quadrilateral that is also cyclic, a line through the midpoint of one side and through the intersection point of the diagonals is perpendicular to the opposite side.

By van Aubel's theorem, if squares are constructed externally on the sides of a quadrilateral, the line segments connecting the centers of opposite squares are perpendicular and equal in length.

Lines in three dimensions[edit]

Up to three lines in three-dimensional space can be pairwise perpendicular, as exemplified by the x, y, and z axes of a three-dimensional Cartesian coordinate system.

In geometry, Brahmagupta's theorem states that if a cyclic quadrilateral is orthodiagonal (that is, has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.[1] It is named after the Indian mathematician Brahmagupta.[2]

More specifically, let A, B, C and D be four points on a circle such that the lines AC and BD are perpendicular. Denote the intersection of AC and BD by M. Drop the perpendicular from M to the line BC, calling the intersection E. Let F be the intersection of the line EM and the edge AD. Then, the theorem states that F is the midpoint AD.

Proof[edit]

Proof of the theorem.

We need to prove that AF = FD. We will prove that both AF and FD are in fact equal to FM.

To prove that AF = FM, first note that the angles FAM and CBM are equal, because they are inscribed angles that intercept the same arc of the circle. Furthermore, the angles CBM and CME are both complementary to angle BCM (i.e., they add up to 90°), and are therefore equal. Finally, the angles CME and FMA are the same. Hence, AFM is an isosceles triangle, and thus the sides AF and FM are equal.

The proof that FD = FM goes similarly: the angles FDM, BCM, BME and DMF are all equal, so DFM is an isosceles triangle, so FD = FM. It follows that AF = FD, as the theorem claims.

See also[edit]

Brahmagupta's formula for the area of a cyclic quadrilateral

QMRIn Euclidean geometry, Brahmagupta's formula finds the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides.

Formula[edit]

Brahmagupta's formula gives the area K of a cyclic quadrilateral whose sides have lengths a, b, c, d as

K=\sqrt{(s-a)(s-b)(s-c)(s-d)}

where s, the semiperimeter, is defined to be

s=\frac{a+b+c+d}{2}.

This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.

If the semiperimeter is not used, Brahmagupta's formula is

K=\frac{1}{4}\sqrt{(-a+b+c+d)(a-b+c+d)(a+b-c+d)(a+b+c-d)}.

Another equivalent version is

K=\frac{\sqrt{(a^2+b^2+c^2+d^2)^2+8abcd-2(a^4+b^4+c^4+d^4)}}{4}\cdot

Proof[edit]

Diagram for reference

Trigonometric proof[edit]

Here the notations in the figure to the right are used. The area K of the cyclic quadrilateral equals the sum of the areas of △ADB and △BDC:

= \frac{1}{2}pq\sin A + \frac{1}{2}rs\sin C.

But since ABCD is a cyclic quadrilateral, ∠DAB = 180° − ∠DCB. Hence sin A = sin C. Therefore,

K = \frac{1}{2}pq\sin A + \frac{1}{2}rs\sin A

K^2 = \frac{1}{4} (pq + rs)^2 \sin^2 A

4K^2 = (pq + rs)^2 (1 - \cos^2 A) = (pq + rs)^2 - (pq + rs)^2 \cos^2 A.\,

Solving for common side DB, in △ADB and △BDC, the law of cosines gives

p^2 + q^2 - 2pq\cos A = r^2 + s^2 - 2rs\cos C. \,

Substituting cos C = −cos A (since angles A and C are supplementary) and rearranging, we have

2 (pq + rs) \cos A = p^2 + q^2 - r^2 - s^2. \,

Substituting this in the equation for the area,

4K^2 = (pq + rs)^2 - \frac{1}{4}(p^2 + q^2 - r^2 - s^2)^2

16K^2 = 4(pq + rs)^2 - (p^2 + q^2 - r^2 - s^2)^2.

The right-hand side is of the form a2 − b2 = (a − b)(a + b) and hence can be written as

[2(pq + rs) - p^2 - q^2 + r^2 +s^2][2(pq + rs) + p^2 + q^2 -r^2 - s^2] \,

which, upon rearranging the terms in the square brackets, yields

= [ (r+s)^2 - (p-q)^2 ][ (p+q)^2 - (r-s)^2 ] \,

= (q+r+s-p)(p+r+s-q)(p+q+s-r)(p+q+r-s). \,

Introducing the semiperimeter S =

p + q + r + s

16K^2 = 16(S-p)(S-q)(S-r)(S-s). \,

Taking the square root, we get

K = \sqrt{(S-p)(S-q)(S-r)(S-s)}.

Non-trigonometric proof[edit]

An alternative, non-trigonometric proof utilizes two applications of Heron's triangle area formula on similar triangles.[1]

Extension to non-cyclic quadrilaterals[edit]

In the case of non-cyclic quadrilaterals, Brahmagupta's formula can be extended by considering the measures of two opposite angles of the quadrilateral:

K=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\theta}

where θ is half the sum of any two opposite angles. (The choice of which pair of opposite angles is irrelevant: if the other two angles are taken, half their sum is 180° − θ. Since cos(180° − θ) = −cos θ, we have cos2(180° − θ) = cos2 θ.) This more general formula is known as Bretschneider's formula.

It is a property of cyclic quadrilaterals (and ultimately of inscribed angles) that opposite angles of a quadrilateral sum to 180°. Consequently, in the case of an inscribed quadrilateral, θ is 90°, whence the term

abcd\cos^2\theta=abcd\cos^2 \left(90^\circ\right)=abcd\cdot0=0, \,

giving the basic form of Brahmagupta's formula. It follows from the latter equation that the area of a cyclic quadrilateral is the maximum possible area for any quadrilateral with the given side lengths.

A related formula, which was proved by Coolidge, also gives the area of a general convex quadrilateral. It is[2]

K=\sqrt{(s-a)(s-b)(s-c)(s-d)-\textstyle{1\over4}(ac+bd+pq)(ac+bd-pq)}\,

where p and q are the lengths of the diagonals of the quadrilateral. In a cyclic quadrilateral, pq = ac + bd according to Ptolemy's theorem, and the formula of Coolidge reduces to Brahmagupta's formula.

Related theorems[edit]

Heron's formula for the area of a triangle is the special case obtained by taking d = 0.

The relationship between the general and extended form of Brahmagupta's formula is similar to how the law of cosines extends the Pythagorean theorem.

QMR In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

The word cyclic is from the Ancient Greek κύκλος (kuklos) which means "circle" or "wheel".

All triangles have a circumcircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be cyclic is a non-square rhombus. The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to have a circumcircle.

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

The word cyclic is from the Ancient Greek κύκλος (kuklos) which means "circle" or "wheel".

Special cases[edit]

Any square, rectangle, isosceles trapezoid, or antiparallelogram is cyclic. A kite is cyclic if and only if it has two right angles. A bicentric quadrilateral is a cyclic quadrilateral that is also tangential and an ex-bicentric quadrilateral is a cyclic quadrilateral that is also ex-tangential.

Characterizations[edit]

A cyclic quadrilateral ABCD

A convex quadrilateral is cyclic if and only if the four perpendicular bisectors to the sides are concurrent. This common point is the circumcenter.[1]

A convex quadrilateral ABCD is cyclic if and only if its opposite angles are supplementary, that is[1]

\alpha + \gamma = \beta + \delta = \pi = 180^{\circ}.

The direct theorem was Proposition 22 in Book 3 of Euclid's Elements.[2] Equivalently, a convex quadrilateral is cyclic if and only if each exterior angle is equal to the opposite interior angle.

Another necessary and sufficient condition for a convex quadrilateral ABCD to be cyclic is that an angle between a side and a diagonal is equal to the angle between the opposite side and the other diagonal.[3] That is, for example,

\angle ACB = \angle ADB.

Ptolemy's theorem expresses the product of the lengths of the two diagonals e and f of a cyclic quadrilateral as equal to the sum of the products of opposite sides:[4]:p.25

\displaystyle ef = ac + bd.

The converse is also true. That is, if this equation is satisfied in a convex quadrilateral, then it is a cyclic quadrilateral.

If two lines, one containing segment AC and the other containing segment BD, intersect at X, then the four points A, B, C, D are concyclic if and only if[5]

\displaystyle AX\cdot XC = BX\cdot XD.

The intersection X may be internal or external to the circle. In the former case, the cyclic quadrilateral is ABCD, and in the latter case, the cyclic quadrilateral is ABDC. When the intersection is internal, the equality states that the product of the segment lengths into which X divides one diagonal equals that of the other diagonal. This is known as the intersecting chords theorem since the diagonals of the cyclic quadrilateral are chords of the circumcircle.

Yet another characterization is that a convex quadrilateral ABCD is cyclic if and only if[6]

\tan{\frac{\alpha}{2}}\tan{\frac{\gamma}{2}}=\tan{\frac{\beta}{2}}\tan{\frac{\delta}{2}}=1.

Area[edit]

The area K of a cyclic quadrilateral with sides a, b, c, d is given by Brahmagupta's formula[4]:p.24

K=\sqrt{(s-a)(s-b)(s-c)(s-d)} \,

where s, the semiperimeter, is s =

(a + b + c + d). It is a corollary to Bretschneider's formula since opposite angles are supplementary. If also d = 0, the cyclic quadrilateral becomes a triangle and the formula is reduced to Heron's formula.

The cyclic quadrilateral has maximal area among all quadrilaterals having the same sequence of side lengths. This is another corollary to Bretschneider's formula. It can also be proved using calculus.[7]

Four unequal lengths, each less than the sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals,[8] which by Brahmagupta's formula all have the same area. Specifically, for sides a, b, c, and d, side a could be opposite any of side b, side c, or side d.

The area of a cyclic quadrilateral with successive sides a, b, c, d and angle B between sides a and b can be expressed as[4]:p.25

K = \tfrac{1}{2}(ab+cd)\sin{B}

or[4]:p.26

K = \tfrac{1}{2}(ac+bd)\sin{\theta}

where θ is either angle between the diagonals. Provided A is not a right angle, the area can also be expressed as[4]:p.26

K = \tfrac{1}{4}(a^2-b^2-c^2+d^2)\tan{A}.

Another formula is[9]:p.83

\displaystyle K=2R^2\sin{A}\sin{B}\sin{\theta}

where R is the radius of the circumcircle. As a direct consequence,[10]

K\le 2R^2

where there is equality if and only if the quadrilateral is a square.

Diagonals[edit]

In a cyclic quadrilateral with successive vertices A, B, C, D and sides a = AB, b = BC, c = CD, and d = DA, the lengths of the diagonals p = AC and q = BD can be expressed in terms of the sides as[4]:p.25,[11][12]:p. 84

p = \sqrt{\frac{(ac+bd)(ad+bc)}{ab+cd}} and q = \sqrt{\frac{(ac+bd)(ab+cd)}{ad+bc}}

so showing Ptolemy's theorem

pq = ac+bd.

According to Ptolemy's second theorem,[4]:p.25,[11]

\frac {p}{q}= \frac{ad+bc}{ab+cd}

using the same notations as above.

For the sum of the diagonals we have the inequality[13]

p+q\ge 2\sqrt{ac+bd}.

Equality holds if and only if the diagonals have equal length, which can be proved using the AM-GM inequality.

Moreover,[14]:p.64,#1639

(p+q)^2 \leq (a+c)^2+(b+d)^2.

In any convex quadrilateral, the two diagonals together partition the quadrilateral into four triangles; in a cyclic quadrilateral, opposite pairs of these four triangles are similar to each other.

If M and N are the midpoints of the diagonals AC and BD, then[15]

\frac{MN}{EF}=\frac{1}{2}\left |\frac{AC}{BD}-\frac{BD}{AC}\right|

where E and F are the intersection points of the extensions of opposite sides.

If ABCD is a cyclic quadrilateral where AC meets BD at E, then[16]

\frac{AE}{CE}=\frac{AB}{CB}\cdot\frac{AD}{CD}.

A set of sides that can form a cyclic quadrilateral can be arranged in any of three distinct sequences each of which can form a cyclic quadrilateral of the same area in the same circumcircle (the areas being the same according to Brahmagupta's area formula). Any two of these cyclic quadrilaterals have one diagonal length in common.[12]:p. 84

Angle formulas[edit]

For a cyclic quadrilateral with successive sides a, b, c, d, semiperimeter s, and angle A between sides a and d, the trigonometric functions of A are given by[17]

\cos A = \frac{a^2 + d^2 - b^2 - c^2}{2(ad + bc)},

\sin A = \frac{2\sqrt{(s-a)(s-b)(s-c)(s-d)}}{(ad+bc)},

\tan \frac{A}{2} = \sqrt{\frac{(s-a)(s-d)}{(s-b)(s-c)}}.

The angle θ between the diagonals satisfies[4]:p.26

\tan \frac{\theta}{2} = \sqrt{\frac{(s-b)(s-d)}{(s-a)(s-c)}}.

If the extensions of opposite sides a and c intersect at an angle φ, then

\cos{\frac{\varphi}{2}}=\sqrt{\frac{(s-b)(s-d)(b+d)^2}{(ab+cd)(ad+bc)}}

where s is the semiperimeter.[4]:p.31

Parameshvara's formula[edit]

A cyclic quadrilateral with successive sides a, b, c, d and semiperimeter s has the circumradius (the radius of the circumcircle) given by[11][18]

R=\frac{1}{4} \sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}.

This was derived by the Indian mathematician Vatasseri Parameshvara in the 15th century.

Using Brahmagupta's formula, Parameshvara's formula can be restated as

4KR=\sqrt{(ab+cd)(ac+bd)(ad+bc)}

where K is the area of the cyclic quadrilateral

Anticenter and collinearities[edit]

Four line segments, each perpendicular to one side of a cyclic quadrilateral and passing through the opposite side's midpoint, are concurrent.[19]:p.131;[20] These line segments are called the maltitudes,[21] which is an abbreviation for midpoint altitude. Their common point is called the anticenter. It has the property of being the reflection of the circumcenter in the "vertex centroid". Thus in a cyclic quadrilateral, the circumcenter, the "vertex centroid", and the anticenter are collinear.[20]

If the diagonals of a cyclic quadrilateral intersect at P, and the midpoints of the diagonals are M and N, then the anticenter of the quadrilateral is the orthocenter of triangle MNP. The vertex centroid is the midpoint of the line segment joining the midpoints of the diagonals.[20]

In a cyclic quadrilateral, the "area centroid" Ga, the "vertex centroid" Gv, and the intersection P of the diagonals are collinear. The distances between these points satisfy[22]

PG_a = \tfrac{4}{3}PG_v.

Other properties[edit]

Japanese theorem

In a cyclic quadrilateral ABCD, the incenters in triangles ABC, BCD, CDA, and DAB are the vertices of a rectangle. This is one of the theorems known as the Japanese theorem. The orthocenters of the same four triangles are the vertices of a quadrilateral congruent to ABCD, and the centroids in those four triangles are vertices of another cyclic quadrilateral.[3]

In a cyclic quadrilateral ABCD with circumcenter O, let P be the point where the diagonals AC and BD intersect. Then angle APB is the arithmetic mean of the angles AOB and COD. This is a direct consequence of the inscribed angle theorem and the exterior angle theorem.

There are no cyclic quadrilaterals with rational area and with unequal rational sides in either arithmetic or geometric progression.[23]

If a cyclic quadrilateral has side lengths that form an arithmetic progression the quadrilateral is also ex-bicentric.

If the opposite sides of a cyclic quadrilateral are extended to meet at E and F, then the internal angle bisectors of the angles at E and F are perpendicular.[8]

Brahmagupta quadrilaterals[edit]

A Brahmagupta quadrilateral[24] is a cyclic quadrilateral with integer sides, integer diagonals, and integer area. All Brahmagupta quadrilaterals with sides a, b, c, d, diagonals e, f, area K, and circumradius R can be obtained by clearing denominators from the following expressions involving rational parameters t, u, and v:

a=[t(u+v)+(1-uv)][u+v-t(1-uv)]

b=(1+u^2)(v-t)(1+tv)

c=t(1+u^2)(1+v^2)

d=(1+v^2)(u-t)(1+tu)

e=u(1+t^2)(1+v^2)

f=v(1+t^2)(1+u^2)

K=uv[2t(1-uv)-(u+v)(1-t^2)][2(u+v)t+(1-uv)(1-t^2)]

4R=(1+u^2)(1+v^2)(1+t^2).

Properties of cyclic quadrilaterals that are also orthodiagonal[edit]

Circumradius and area[edit]

For a cyclic quadrilateral that is also orthodiagonal (has perpendicular diagonals), suppose the intersection of the diagonals divides one diagonal into segments of lengths p1 and p2 and divides the other diagonal into segments of lengths q1 and q2. Then[25] (the first equality is Proposition 11 in Archimedes' Book of Lemmas)

D^2=p_1^2+p_2^2+q_1^2+q_2^2=a^2+c^2=b^2+d^2

where D is the diameter of the circumcircle. This holds because the diagonals are perpendicular chords of a circle. These equations imply that the circumradius R can be expressed as

R=\tfrac{1}{2}\sqrt{p_1^2+p_2^2+q_1^2+q_2^2}

or, in terms of the sides of the quadrilateral, as

R=\tfrac{1}{2}\sqrt{a^2+c^2}=\tfrac{1}{2}\sqrt{b^2+d^2}.

It also follows that

a^2+b^2+c^2+d^2=8R^2.

Thus, according to Euler's quadrilateral theorem, the circumradius can be expressed in terms of the diagonals p and q, and the distance x between the midpoints of the diagonals as

R=\sqrt{\frac{p^2+q^2+4x^2}{8}}.

A formula for the area K of a cyclic orthodiagonal quadrilateral in terms of the four sides is obtained directly when combining Ptolemy's theorem and the formula for the area of an orthodiagonal quadrilateral. The result is

K=\tfrac{1}{2}(ac+bd).

Other properties[edit]

In a cyclic orthodiagonal quadrilateral, the anticenter coincides with the point where the diagonals intersect.[19]

Brahmagupta's theorem states that for a cyclic quadrilateral that is also orthodiagonal, the perpendicular from any side through the point of intersection of the diagonals bisects the opposite side.[19]

If a cyclic quadrilateral is also orthodiagonal, the distance from the circumcenter to any side equals half the length of the opposite side.[19]

In a cyclic orthodiagonal quadrilateral, the distance between the midpoints of the diagonals equals the distance between the circumcenter and the point where the diagonals intersect.[19]

Other properties[edit]

In a cyclic orthodiagonal quadrilateral, the anticenter coincides with the point where the diagonals intersect.[19]

If a cyclic quadrilateral is also orthodiagonal, the distance from the circumcenter to any side equals half the length of the opposite side.[19]

In a cyclic orthodiagonal quadrilateral, the distance between the midpoints of the diagonals equals the distance between the circumcenter and the point where the diagonals intersect.[19]

The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:[1]:p. 78

Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly. Then M is the midpoint of XY.

Contents [hide]

1 Proof

2 History

3 References

4 Bibliography

5 External links

Proof[edit]

A formal proof of the theorem is as follows: Let the perpendiculars XX′ and XX″ be dropped from the point X on the straight lines AM and DM respectively. Similarly, let YY′ and YY″ be dropped from the point Y perpendicular to the straight lines BM and CM respectively.

Proof of Butterfly theorem

Now, since

\triangle MXX' \sim \triangle MYY',\,

{MX \over MY} = {XX' \over YY'},

\triangle MXX'' \sim \triangle MYY'',\,

{MX \over MY} = {XX'' \over YY''},

\triangle AXX' \sim \triangle CYY'',\,

{XX' \over YY''} = {AX \over CY},

\triangle DXX'' \sim \triangle BYY',\,

{XX'' \over YY'} = {DX \over BY},

From the preceding equations, it can be easily seen that

\left({MX \over MY}\right)^2 = {XX' \over YY' } {XX'' \over YY''},

{} = {AX.DX \over CY.BY},

{} = {PX.QX \over PY.QY},

{} = {(PM-XM).(MQ+XM) \over (PM+MY).(QM-MY)},

{} = { (PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2},

since PM = MQ.

Now,

{ (MX)^2 \over (MY)^2} = {(PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}.

So, it can be concluded that MX = MY, or M is the midpoint of XY.

An alternate proof can be found using projective geometry.[2]

History[edit]

Proving the butterfly theorem was posed as a problem by William Wallace in The Gentlemen's Mathematical Companion (1803). Three solutions were published in 1804, and in 1805 Sir William Herschel posed the question again in a letter to Wallace. Rev. Thomas Scurr asked the same question again in 1814 in the Gentlemen's Diary or Mathematical Repository.[3]

These proofs involve perpendicular quadrant lines

The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:[1]:p. 78

Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly. Then M is the midpoint of XY.

Contents [hide]

1 Proof

2 History

3 References

4 Bibliography

5 External links

Proof[edit]

Proof of Butterfly theorem

Now, since

\triangle MXX' \sim \triangle MYY',\,

{MX \over MY} = {XX' \over YY'},

\triangle MXX'' \sim \triangle MYY'',\,

{MX \over MY} = {XX'' \over YY''},

\triangle AXX' \sim \triangle CYY'',\,

{XX' \over YY''} = {AX \over CY},

\triangle DXX'' \sim \triangle BYY',\,

{XX'' \over YY'} = {DX \over BY},

From the preceding equations, it can be easily seen that

\left({MX \over MY}\right)^2 = {XX' \over YY' } {XX'' \over YY''},

{} = {AX.DX \over CY.BY},

{} = {PX.QX \over PY.QY},

{} = {(PM-XM).(MQ+XM) \over (PM+MY).(QM-MY)},

{} = { (PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2},

since PM = MQ.

Now,

{ (MX)^2 \over (MY)^2} = {(PM)^2 - (MX)^2 \over (PM)^2 - (MY)^2}.

So, it can be concluded that MX = MY, or M is the midpoint of XY.

An alternate proof can be found using projective geometry.[2]

History[edit]

QMRThe Four Lords of the Warring States were four powerful aristocrats of the late Warring States period of Chinese history who exerted a strong influence on the politics of their respective states in the third century BCE.[1]

During this time, the Zhou king was a mere figurehead, and seven states led by aristocratic families competed for real power. Although they were not themselves monarchs, four aristocrats stood out because of their tremendous military power and wealth: Lord Mengchang (d. 279 bce) of Qi, Lord Xinling (d. 242 bce) of Wei, Lord Pingyuan (d. 251 bce) of Zhao and Lord Chunshen (d. 237 bce) of Chu.[2]

All four were renowned for their activity in the politics of their era as well as being the persona of their state respectively at the time; they also wielded influence via the cultivation and housing of many talented house-guests, who often included learned men and tacticians. As such, they came to be the most prominent patrons of the shi (士) or scholar-knights, stimulating the intellectual life of the time. Their prestige became the inspiration for Lü Buwei when he created his academic analogue in Qin. Lord Mengchang[edit]

Main article: Lord Mengchang

Lord Mengchang was an aristocrat of the State of Qi. He was born Tian Wen, son of Tian Ying and the grandson of King Wei of Qi. He succeeded his father's fief in Xue.

Lord Xinling[edit]

Main article: Lord Xinling

Born as Wei Wuji, he was the son of King Zhao of the State of Wei and younger half-brother to King Anxi of Wei. In 277 bce, King Anxi assigned Wei Wuji the fief of Xinling.

At the height of his career, he was the supreme commander of the armed forces of the Kingdom of Wei. After stepping down, Lord Xinling became dispirited and died in 243 bce.

Lord Pingyuan[edit]

Main article: Lord Pingyuan

Born Zhao Sheng, he was a son of King Wuling of Zhao, brother of King Huiwen and uncle to King Xiaocheng. During his life, he was thrice appointed the Prime Minister of the State of Zhao.

Zhao Sheng's fief was the City of Dongwu. Lord Pingyuan was his title, and some of his famous retainers included the philosophers Xun Kuang and Gongsun Long, the Yin and Yang master Zou Yan, and the diplomat Mao Sui.

Lord Chunshen[edit]

Main article: Lord Chunshen

Born Huang Xie, he was originally a government official working for King Qingxiang of Chu, and later followed Crown Prince Wan when he spent ten years as a hostage in the Kingdom of Qin.

After the death of King Qingxiang, Prince Wan and Huang Xie returned to the Kingdom of Chu. Prince Wan was enthroned as King Kaolie of Chu, while Huang Xie was appointed Prime Minister and received the title of Lord Chunshen. For the next 25 years, Lord Chunshen remained Prime Minister of Chu, until his assassination by Li Yuan in 238 bce.

These four lords are paralleled in some books of the Records of the Grand Historian, the first of the Twenty-Four Histories of China.

In the Biographies of Lord Pingyuan and Yu Qing,[3]

At this time, [in addition to Lord Pingyuan in Zhao,] in Qi lived Mengchang, in Wei Xinling, and in Chu Chunshen. They competed to invite shi (talents).

In the Biography of Lord Chunshen,[4]

Lord Chunshen now stood as the prime minister of the Kingdom of Chu. At this time, in Qi lived Lord Mengchang, in Zhao Lord Pingyuan, and in Wei Lord Xinling. They competed to humble themselves before shi (talents) [to hire them], invited brilliant guests, and tried to defeat each other. They sustained their states and held the real power.

QMRThe Four Modernizations were goals first set forth by Zhou Enlai in 1963, and enacted by Deng Xiaoping, starting in 1978, to strengthen the fields of agriculture, industry, national defense, and science and technology in China.[1] The Four Modernizations were adopted as a means of rejuvenating China's economy in 1978 following the death of Mao Zedong, and were among the defining features of Deng Xiaoping's tenure as head of the party

They were introduced as early as January 1963: at the Conference on Scientific and Technological Work held in Shanghai that month, Zhou Enlai called for professionals in the sciences to realize "the Four Modernizations."[2] In February 1963, at the National Conference on Agricultural Science and Technology Work, Nie Rongzhen specifically referred to the Four Modernizations as comprising agriculture, industry, national defense, and science and technology.[3] In 1975, in one of his last public acts, Zhou Enlai made another pitch for the Four Modernizations at the 4th National People's Congress. After Zhou's death and Mao’s soon thereafter, Deng Xiaoping assumed control of the party in late 1978. In December 1978 at the Third Plenum of the 11th Central Committee, Deng Xiaoping announced the official launch of the Four Modernizations, formally marking the beginning of the reform era.

The science and technology modernization although understood by Chinese leaders as being key to the transformation of industry and the economy, proved to be more of a theoretical goal versus an achievable objective. This was primarily due to decades-long isolation of Chinese scientists from the western international community, outmoded universities, and an overall lack of access to advanced scientific equipment, information technology, and management knowhow. Recognizing the need for technical assistance to spur this most important modernization, the Chinese Government elicited the support of the United Nations Development Programme (UNDP) in the fall of 1978 to scope out and provide financial resources for the implementation of an initial complement of targeted projects. The initial projects from 1979–1984 included the establishment of overseas on-the-job training and academic programs, set-up of information processing centers at key government units, and the development of methods to make informed decisions within the Chinese context based on market principles. The key advisor to the Chinese Government on behalf of the UNDP was Jack Fensterstock of the United States. This first technical assistance effort (CPR/79-001) by the UNDP led to the entry of large-scale multilateral funding agencies including the World Bank and the Asian Development Bank.

The Four Modernizations were designed to make China a great economic power by the early 21st century. These reforms essentially stressed economic self-reliance. The People's Republic of China decided to accelerate the modernization process by stepping up the volume of foreign trade by opening up its markets, especially the purchase of machinery from Japan and the West. By participating in such export-led growth, China was able to speed up its economic development through foreign investment, a more open market, access to advanced technologies, and management experience.

Controversy[edit]

On December 5, 1978 in Beijing, former red guard Wei Jingsheng posted on the Democracy Wall the Fifth Modernization as being "democracy". He was arrested a few months later and jailed for 15 years.[4]

QMRDeng Xiaoping Theory (simplified Chinese: 邓小平理论; traditional Chinese: 鄧小平理論; pinyin: Dèng Xiǎopíng Lǐlùn), also known as Dengism, is the series of political and economic ideologies first developed by Chinese leader Deng Xiaoping. The theory does not reject Marxism–Leninism or Mao Zedong Thought but instead seeks to adapt them to the existing socio-economic conditions of China.

China's economic growth largely owes its success to this pragmatism of Deng Xiaoping's theory. The task faced by Deng was twofold: to promote modernization while preserving the ideological unity of the CPC and its control of the difficult process of reforms.

It was generalized by the concept of the Four Modernizations.

This became the main motivation for ideological conservatism of Deng Xiaoping Theory: "Four Cardinal Principles" which the Communist Party must uphold, namely,

Upholding the basic spirit of Communism

Upholding the People's democratic dictatorship political system

Upholding the leadership of the Communist Party

Upholding Marxism–Leninism and Mao Zedong Thought

QMRThe Han dynasty formally recognized four sources of law: lü (律: "codified laws"), ling (令: "the emperor's order"), ke (科: "statutes inherited from previous dynasties") and bi (比: "precedents"), among which ling has the highest binding power over the other three. Most legal professionals were not lawyers but generalists trained in philosophy and literature. The local, classically trained, Confucian gentry played a crucial role as arbiters and handled all but the most serious local disputes.

QMRClassical Chinese is distinguished from written vernacular Chinese in its style, which appears extremely concise and compact to modern Chinese speakers, and to some extent in the use of different lexical items (vocabulary). An essay in Classical Chinese, for example, might use half as many Chinese characters as in vernacular Chinese to relate the same content.

In terms of conciseness and compactness, Classical Chinese rarely uses words composed of two Chinese characters; nearly all words are of one syllable only. This stands directly in contrast with modern Northern Chinese varieties including Mandarin, in which two-syllable, three-syllable, and four-syllable words are extremely common, whilst although two-syllable words are also quite common within modern Southern Chinese varieties, they are still more archaic in that they use more one-syllable words than Northern Chinese varieties. This phenomenon exists, in part, because polysyllabic words evolved in Chinese to disambiguate homophones that result from sound changes. This is similar to such phenomena in English as the pen–pin merger of many dialects in the American south: because the words "pin" and "pen" sound alike in such dialects of English, a certain degree of confusion can occur unless one adds qualifiers like "ink pen" and "stick pin." Similarly, Chinese has acquired many polysyllabic words in order to disambiguate monosyllabic words that sounded different in earlier forms of Chinese but identical in one region or another during later periods. Because Classical Chinese is based on the literary examples of ancient Chinese literature, it has almost none of the two-syllable words present in modern Chinese varieties.

Business management financial market tenets these are Warren Buffets four groups of tenets.

Buffet way four principals

QMRThe 24 Game is an arithmetical card game in which the objective is to find a way to manipulate four integers so that the end result is 24. For example, for the card with the numbers 4, 7, 8, 8, a possible solution is (7-(8\div8))\times4=24.

The game has been played in Shanghai since the 1960s,[citation needed] using playing cards. It is similar to the card game Maths24.

Mental arithmetic and fast thinking are necessary skills for competitive play. Pencil and paper will slow down a player, and are generally not allowed during play anyway.

In the original version of the game played with a standard 52-card deck, there are \tbinom {4+13-1}4=1820 four-card combinations.[citation needed]

Additional operations, such as square root and factorial, allow more possible solutions to the game. For instance, a set of 1,1,1,1 would be impossible to solve with only the five basic operations. However, with the use of factorials, it is possible to get 24 as (1+1+1+1)!=24.

The game has been played in Shanghai since the 1960s,[citation needed] using playing cards. It is similar to the card game Maths24.

QMRMaths24 is a competitive, arithmetical card game aimed predominantly at primary and high school pupils. Although it can be played informally, the game was organised and operated within Southern Africa in a series of interschool, geographically increasing tournaments. The game experienced its peak during the 1990s, and is now no longer produced or played in any official manner.

History[edit]

The Maths24 game was devised, sponsored and officiated by the Old Mutual insurance company. It first began appearing in schools in 1989, and was even sold commercially to the public for a short period in the early 1990s.

Old Mutual devised the game and competitions chiefly as a promotional activity to publicise the company. The project aimed to introduce pupils and encourage their participation in mathematics via entertaining activities.

Interest in Maths24 began to taper off towards the end of the 1990s. Although official sponsorship and promotion of the game stopped, many schools and individuals continue to play the game informally.

Due to its unconventional release and propagation, the game went through a variety of rules, playing-styles and even names, being known for some time as The I-Got-It Game.

Card Description[edit]

The cards are double-sided, thin cardboard squares with sides measuring approximately 10 cm. The conventional cards bare the Old Mutual logo in the centres of each side, however with the green and white inverted on one side to differentiate. Later variations of the cards bore red, blue and black logos.

The conventional card displays four numbers, each a single digit from 1 to 9. Numbers may repeat. The cards are designed to be viewed from any angle.

The card difficulty is ranked by displaying one, two or three dots in each corner of the card in white, red and yellow respectively, as the difficulty increases.

Rules[edit]

Although official rules were later published, the game evolved with common basic rules, and many smaller variations.

Any number of competitors (usually four at most) sit around a table. The cards are placed, one at a time, in the centre of the table. The first person to cover the card with their hand and claim to have the solution would then be given the first opportunity to give their answer.

Cards are solved by using the numbers, applying only the addition, subtraction, multiplication and division operations to achieve a final mathematical solution of 24.

All four numbers must be included. The numbers can only be used once. No other mathematical operations are allowed. There may be more than one way to solve each card.

Tournament Play[edit]

The Maths24 game was intended and mostly played in tournament scenarios, ranging from school to international levels.

Competitors are initially distributed into tables of four, each with their own adjudicator. The game is played in rounds, with participants competing for points. Points are earned by solving cards, with one, two or three points assigned to cards of increasing difficulty. After claiming a solution to a card, if a participant failed to give a correct or legal solution their points would be deducted according to the difficulty. Rounds continue for a predefined length of time, at the end of which the points are tallied and the winners proceed.

Participants claim a card by covering it with their hand after it has been placed onto the table, and audibly declaring "I've got it". Later, due to minor injuries, the rules enforced covering only the centre of the card with only the index and middle fingers.

At the top competition levels, there would typically be almost no delay between a card's placement and a participant's claiming it. This is because there are only a finite number of cards that can be made, and participant's ability to give a solution from memory. Also, participants could exploit the small delay, instead, between claiming a card and giving its solution, to work out the solution then.

Variations[edit]

Because of the finite extent of the basic cards, many variations and adaptations of the game were introduced to add complexity to tournaments.

Combinations[edit]

One[1] of the officially recognized, international top-twenty Maths24 participants has shown that although the total possible number of four repeatable single-digit combinations, where order does not matter is 495, there are only 403 legally solvable possible combinations.

All 403 possible cards, in ascending order, with one possible solution are given below:

QMRThe 2010 census put Shanghai's total population at 23,019,148, a growth of 37.53% from 16,737,734 in 2000.[82][83] 20.6 million of the total population, or 89.3%, are urban, and 2.5 million (10.7％) are rural.[84] Based on the population of its total administrative area, Shanghai is the second largest of the four direct-controlled municipalities of China, behind Chongqing, but is generally considered the largest Chinese city because Chongqing's urban population is much smaller

Shanghai has four major railway stations: Shanghai Railway Station, Shanghai South Railway Station, Shanghai West Railway Station, and Shanghai Hongqiao Railway Station. Three are connected to the metro network and serve as hubs in the railway network of China. Two main railways terminate in Shanghai: Jinghu Railway from Beijing, and Huhang Railway from Hangzhou. Hongqiao Station also serves as the main Shanghai terminus of three high-speed rail lines: the Shanghai–Hangzhou High-Speed Railway, the Shanghai–Nanjing High-Speed Railway, and the Beijing–Shanghai High-Speed Railway.

QMRPell's equation (also called the Pell–Fermat equation) is any Diophantine equation of the form

x^2-ny^2=1\,

where n is a given positive nonsquare integer and integer solutions are sought for x and y. In Cartesian coordinates, the equation has the form of a hyperbola; solutions occur wherever the curve passes through a point whose x and y coordinates are both integers, such as the trivial solution with x = 1 and y = 0. Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y.

Recall that to square something is to make it a quadrant

QMRPell's equation, first misnamed by Euler.[53] He wrote on the link between continued fractions and Pell's equation.[54]

First steps towards analytic number theory. In his work of sums of four squares, partitions, pentagonal numbers, and the distribution of prime numbers, Euler pioneered the use of what can be seen as analysis (in particular, infinite series) in number theory. Since he lived before the development of complex analysis, most of his work is restricted to the formal manipulation of power series. He did, however, do some very notable (though not fully rigorous) early work on what would later be called the Riemann zeta function.[55]

QMRIn number theory, quadratic integers are a generalization of the integers to quadratic fields. Quadratic integers are the solutions of equations of the form

x2 + Bx + C = 0

with B and C integers. They are thus algebraic integers of the degree two. When algebraic integers are considered, usual integers are often called rational integers.

Common examples of quadratic integers are the square roots of integers, such as √2, and the complex number i = √–1, which generates the Gaussian integers. Another common example is the non-real cubic root of unity

-1 + √–3

, which generates the Eisenstein integers.

Quadratic integers occur in the solutions of many Diophantine equations, such as Pell's equations. The study of rings of quadratic integers is basic for many questions of algebraic number theory.

Medieval Indian mathematicians had already discovered a multiplication of quadratic integers of the same D, which allowed them to solve some cases of Pell's equation.[citation needed]

The characterization[clarification needed] of the quadratic integers was first given by Richard Dedekind in 1871.[1][2]

Definition[edit]

A quadratic integer is a complex number which is a solution of an equation of the form

x2 + Bx + C = 0

with B and C integers. In other words, a quadratic integer is an algebraic integer in a quadratic field. Each quadratic integer that is not an integer lies in a uniquely determined quadratic field, namely, the extension of \mathbb{Q} generated by the square-root of B2-4C, which can always be written in the form \mathbb{Q}(\sqrt{D}), where D is the unique square-free integer for which B2 – 4C = DE2 for some integer E.

The quadratic integers (including the ordinary integers), which belong to a quadratic fieds \mathbb{Q}(\sqrt{D}), form a integral domain called ring of integers of \mathbb{Q}(\sqrt{D}).

Here and in the following, D is supposed to be a square-free integer. This does not restricts the generality, as the equality √a2D = a√D (for any positive integer a) implies \mathbb{Q}(\sqrt{D})=\mathbb{Q}(\sqrt{a^2D}).

Every quadratic integer may be written a + ωb , where a and b are integers, and where ω is defined by:

\omega =

\begin{cases}

\sqrt{D} & \mbox{if }D \equiv 2, 3 \pmod{4} \\

{{1 + \sqrt{D}} \over 2} & \mbox{if }D \equiv 1 \pmod{4}

\end{cases}

(as D has been supposed square-free the case D \equiv 0\pmod{4} is impossible, since it would imply that D would be divisible by the square 4).

Although the quadratic integers belonging to a given quadratic field form a ring, the set of all quadratic integers is not a ring, because it is not closed under addition, as \sqrt{2}+\sqrt{3} is an algebraic integer, which has a minimal polynomial of degree four.

Norm and conjugation[edit]

A quadratic integer in \mathbb{Q}(\sqrt{D}) may be written

a + b√D,

where either a and b are either integers, or, only if D ≡ 1 (mod 4), halves of odd integers. The norm of such a quadratic integer is

N(a + b√D) = a2 – b2D.

The norm of a quadratic integer is always an integer. If D < 0, the norm of a quadratic integer is the square of its absolute value as a complex number (this is false if D > 0). The norm is a completely multiplicative function, which means that the norm of a product of quadratic integers is always the product of their norms.

Every quadratic integer a + b√D has a conjugate

\overline{a+b\sqrt{D}} = a-b\sqrt{D}.

An algebraic integer has the same norm as its conjugate, and this norm is the product of the algebraic integer and its conjugate. The conjugate of a sum or a product of algebraic integers it the sum or the product (respectively) of the conjugates. This means that the conjugation is an automorphism of the ring of the integers of \mathbb{Q}(\sqrt{D}).

Units[edit]

A quadratic integer is a unit in the ring of the integers of \mathbb{Q}(\sqrt{D}) if and only if its norm is 1 or –1. In the first case its multiplicative inverse is its conjugate. It is the opposite of its conjugate in the second case.

If D < 0, the ring of the integers of \mathbb{Q}(\sqrt{D}) has at most six units. In the case of the Gaussian integers (D = –1), the four units are 1, –1, √–1, –√–1. In the case of the Eisenstein integers (D = –3), the six units are ±1,

±1 ± √–3

. For all other negative D, there are only two units that are 1 and –1.

If D > 0, the ring of the integers of \mathbb{Q}(\sqrt{D}) has infinitely many units that are equal to ±ui, where i is an arbitrary integer, and u is a particular unit called a fundamental unit. Given a fundamental unit u, there are three other fundamental units, its conjugate \overline{u}, and also -u and -\overline{u}. Commonly, one calls the fundamental unit, the unique one which has an absolute value greater than 1 (as a real number). It is the unique fundamental unit that may be written a + b√D, with a and b positive (integers or halves of integers).

The fundamental units for the 10 smallest positive square-free D are 1 + √2, 2 + √3,

1 + √5

(the golden ratio), 5 + 2√6, 8 + 3√7, 3 + √10, 10 + 3√11,

3 + √13

, 15 + 4√14, 4 + √15. For larger D, the coefficients of the fundamental unit may be very large. For example, for D = 19, 31, 43, the fundamental units are respectively 170 + 39 √19, 1520 + 273 √31 and 3482 + 531 √43.

Quadratic integer rings[edit]

Every square-free integer (different of 0 and 1) D defines a quadratic integer ring, which is the integral domain of the algebraic integers contained in \mathbf{Q}(\sqrt{D}). It is the set Z[ω] =a + ωb : a, b ∈ Z, where ω is defined as above. It is called the ring of integers of Q(√D) and often denoted \mathcal{O}_{\mathbf{Q}(\sqrt{D})}. By definition, it is the integral closure of Z in \mathbf{Q}(\sqrt{D}).

The properties of the quadratic integers (and more generally of algebraic integers) has been a long standing problem, which has motivated the elaboration of the notions of ring and ideal. In particular the fundamental theorem of arithmetic is not true in many rings of quadratic integers. However there is a unique factorization for ideals, which is expressed by the fact that every ring of algebraic integers is a Dedekind domain.

Quadratic integer rings and their associated quadratic fields are thus commonly the starting examples of most studies of algebraic number fields.

The quadratic integer rings divide in two classes depending on the sign of D. If D > 0, all elements of \mathcal{O}_{\mathbf{Q}(\sqrt{D})} are real, and the ring is a real quadratic integer ring. If D < 0, the only real elements of \mathcal{O}_{\mathbf{Q}(\sqrt{D})} are the ordinary integers, and the ring is a complex quadratic integer ring.

Examples of complex quadratic integer rings[edit]

Gaussian integers

Eisenstein primes

For D < 0, ω is a complex (imaginary or otherwise non-real) number. Therefore, it is natural to treat a quadratic integer ring as a set of algebraic complex numbers.

A classic example is \mathbf{Z}[\sqrt{-1}], the Gaussian integers, which was introduced by Carl Gauss around 1800 to state his biquadratic reciprocity law.[3]

The elements in \mathcal{O}_{\mathbf{Q}(\sqrt{-3})} = \mathbf{Z}\left[{{1 + \sqrt{-3}} \over 2}\right] are called Eisenstein integers.

Both rings mentioned above are rings of integers of cyclotomic fields Q(ζ4) and Q(ζ3) correspondingly. In contrast, Z[√−3] is not even a Dedekind domain.

Both above examples are principal ideal rings and also Euclidean domains for the norm. This is not the case for

\mathcal{O}_{\mathbf{Q}(\sqrt{-5})} = \mathbf{Z}\left[\sqrt{-5}\right],

which is not even a unique factorization domain. This can be shown as follows.

In \mathcal{O}_{\mathbf{Q}(\sqrt{-5})}, we have

9 = 3\cdot3 = (2+\sqrt{-5})(2-\sqrt{-5}).

The factors 3, 2+\sqrt{-5} and 2-\sqrt{-5} are irreducible, as they have all a norm of 9, and if they were not irreducible, they would have a factor of norm 3, which is impossible, the norm of an element different of ±1 being at least 4. Thus the factorization of 9 into irreducible factors is not unique.

The ideals \langle 3, 1+\sqrt{-5}\rangle and \langle 3, 1-\sqrt{-5}\rangle are not principal, as a simple computation shows that their product is the ideal generated by 3, and, if they were principal, this would imply that 3 would not be irreducible.

Examples of real quadratic integer rings[edit]

Powers of the golden ratio

For D > 0, ω is a positive irrational real number, and the corresponding quadratic integer ring is a set of algebraic real numbers. The solutions of the Pell's equation X2 − D Y2 = 1, a Diophantine equation that has been widely studied, are the units of these rings, for D ≡ 2, 3 (mod 4).

For D = 5, ω =

1+√5

is the golden ratio. This ring was studied by Peter Gustav Lejeune Dirichlet. Its invertible elements have the form ±ωn, where n is an arbitrary integer. This ring also arises from studying 5-fold rotational symmetry on Euclidean plane, for example, Penrose tilings.[4]

Indian mathematician Brahmagupta treated the Pell's equation X2 − 61 Y2 = 1, corresponding to the ring is Z[√61]. Some results were presented to European community by Pierre Fermat in 1657.[which?]

Principal rings of quadratic integers[edit]

Unique factorization property is not always verified for rings of quadratic integers, as seen above for the case of Z[√−5]. However, as for every Dedekind domain, a ring of quadratic integers is a unique factorization domain if and only if it is a principal ideal domain. This occurs if and only if the class number of the corresponding quadratic field is one.

The imaginary rings of quadratic integers that are principal ideal rings have been completely determined. These are \mathcal{O}_{\mathbf{Q}(\sqrt{D})} for

D = −1, −2, −3, −7, −11, −19, −43, −67, −163.

This result was first conjectured by Gauss and proven by Kurt Heegner, although Heegner's proof was not believed until Harold Stark gave a later proof in 1967. (See Stark–Heegner theorem.) This is a special case of the famous class number problem.

There are many known positive integers D > 0, for which the ring of quadratic integers is a principal ideal ring. However, the complete list is not known; it is not even known if the number of these principal ideal rings is finite or not.

Euclidean rings of quadratic integers[edit]

See also: Euclidean domain § Norm-Euclidean fields

When a ring of quadratic integers is a principal ideal domain, it is interesting to know if it is a Euclidean domain. This problem has been completely solved as follows.

Equipped with the norm N(a + b\sqrt{D}) = a^2 - Db^2, as an Euclidean function, \mathcal{O}_{\mathbf{Q}(\sqrt{D})} is an Euclidean domain for negative D when

D = −1, −2, −3, −7, −11, [5]

and, for positive D, when

D = 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 (sequence A048981 in OEIS).

There is no other ring of quadratic integers that is Euclidean with the norm as a Euclidean function.[6]

For negative D, a ring of quadratic integers is Euclidean if and only if the norm is a Euclidean function for it. It follows that, for

D = −19, −43, −67, −163,

the four corresponding rings of quadratic integers are among the rare known examples of principal ideal domains that are not Euclidean domains.

On the other hand, the generalized Riemann hypothesis implies that a ring of real quadratic integers that is a principal ideal domain is also a Euclidean domain for some Euclidean function.[citation needed

QMRIn mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of an odd integer; a mixture of integers and half-integers is excluded). The set of all Hurwitz quaternions is

H = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z} \;\mbox{ or }\, a,b,c,d \in \mathbb{Z} + \tfrac{1}{2}\right\}.

H is closed under quaternion multiplication and addition, which makes it a subring of the ring of all quaternions H. Hurwitz quaternions were introduced by Hurwitz (1919).

A Lipschitz quaternion (or Lipschitz integer) is a quaternion whose components are all integers. The set of all Lipschitz quaternions

L = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z}\right\}

forms a subring of the Hurwitz quaternions H. Hurwitz integers have the advantage over Lipschitz integers that you can perform Euclidean division on them, obtaining a small remainder.

H = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z} \;\mbox{ or }\, a,b,c,d \in \mathbb{Z} + \tfrac{1}{2}\right\}.

H is closed under quaternion multiplication and addition, which makes it a subring of the ring of all quaternions H. Hurwitz quaternions were introduced by Hurwitz (1919).

A Lipschitz quaternion (or Lipschitz integer) is a quaternion whose components are all integers. The set of all Lipschitz quaternions

L = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z}\right\}

forms a subring of the Hurwitz quaternions H. Hurwitz integers have the advantage over Lipschitz integers that you can perform Euclidean division on them, obtaining a small remainder.

QMRA Kummer extension is a field extension L/K, where for some given integer n > 1 we have

K contains n distinct nth roots of unity (i.e., roots of Xn−1)

L/K has abelian Galois group of exponent n.

For example, when n = 2, the first condition is always true if K has characteristic ≠ 2. The Kummer extensions in this case include quadratic extensions L = K(√a) where a in K is a non-square element. By the usual solution of quadratic equations, any extension of degree 2 of K has this form. The Kummer extensions in this case also include biquadratic extensions and more general multiquadratic extensions. When K has characteristic 2, there are no such Kummer extensions.

QMRThe square root of 2, written in mathematics as √2 or 2^{1/2}, is the positive algebraic number that, when multiplied by itself, gives the number 2. Technically, it is called the principal square root of 2, to distinguish it from the negative number with the same property.

Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. Its numerical value, truncated to 65 decimal places, is:

1.41421356237309504880168872420969807856967187537694807317667973799... (sequence A002193 in OEIS).

The square root of 2.

The approximation 99/70 (≈ 1.41429) for the square root of two is frequently used. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. 7.2 × 10−5). The approximation 665857/470832 is valid to within 1.13 x 10−12: its square is 2.0000000000045....

Plato tried to find this by drawing a quadrant

The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of √2 in four sexagesimal figures, 1 24 51 10, which is accurate to about six decimal digits,[1] and is the closest possible three-place sexagesimal representation of √2:

1 + \frac{24}{60} + \frac{51}{60^2} + \frac{10}{60^3} = \frac{30547}{21600} = 1.41421\overline{296}.

Another early close approximation is given in ancient Indian mathematical texts, the Sulbasutras (c. 800–200 BC) as follows: Increase the length [of the side] by its third and this third by its own fourth less the thirty-fourth part of that fourth.[2] That is,

1 + \frac{1}{3} + \frac{1}{3 \cdot 4} - \frac{1}{3 \cdot4 \cdot 34} = \frac{577}{408} = 1.41421\overline{56862745098039}.

This ancient Indian approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, that can be derived from the continued fraction expansion of √2. Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation.

Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational. Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it.[3][4][5] The square root of two is occasionally called "Pythagoras' number" or "Pythagoras' Constant", for example by Conway & Guy (1996).[6]

The Babylonian tablet was a quadrant

QMRComplex coordinates[edit]

The unit square can also be thought of as a subset of the complex plane, the topological space formed by the complex numbers. In this view, the four corners of the unit square are at the four complex numbers 0, 1, i, and 1 + i.

QMRIn geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations (or honeycombs) of Euclidean 3-space and their duals, as three-dimensional analogue of the Platonic, Archimedean, and Catalan tiling of the plane. The singular vertex figure of an architectonic tessellation is the dual of the cell of catoptric tessellation. The cubille is the only Platonic (regular) tessellation of 3-space, and is self-dual.

The pairs of architectonic and catoptric tessellations are listed below with their symmetry group. These tessellations only represent four symmetry space groups, and also all within the cubic crystal system. Many of these tessellations can be defined in multiple symmetry groups, so in each case the highest symmetry is expressed.

These four symmetry groups are labeled as:

Label Description space group

Intl symbol Geometric

notation[2] Coxeter

notation Fibrifold

notation

bc bicubic symmetry

or extended cubic symmetry (221) Im3m I43 [[4,3,4]]

CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c1.png 8°:2

nc normal cubic symmetry (229) Pm3m P43 [4,3,4]

CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 4−:2

fc half-cubic symmetry (225) Fm3m F43 [4,31,1] = [4,3,4,1+]

CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png 2−:2

d diamond symmetry

or extended quarter-cubic symmetry (227) Fd3m Fd4n3 [[3[4]]] = [[1+,4,3,4,1+]]

CDel branch c1.pngCDel 3ab.pngCDel branch c2.png 2+:2

Bonding[edit]

Molecules, by definition, are most often held together with covalent bonds involving single, double, and/or triple bonds, where a "bond" is a shared pair of electrons (the other method of bonding between atoms is called ionic bonding and involves a positive cation and a negative anion).

Molecular geometries can be specified in terms of bond lengths, bond angles and torsional angles. The bond length is defined to be the average distance between the nuclei of two atoms bonded together in any given molecule. A bond angle is the angle formed between three atoms across at least two bonds. For four atoms bonded together in a chain, the torsional angle is the angle between the plane formed by the first three atoms and the plane formed by the last three atoms.

There exists a mathematical relationship among the bond angles for one central atom and four peripheral atoms (labeled 1 through 4) expressed by the following determinant. This constraint removes one degree of freedom from the choices of (originally) six free bond angles to leave only five choices of bond angles. (Note that the angles \theta_{11}, \theta_{22}, \theta_{33}, and \theta_{44} are always zero and that this relationship can be modified for a different number of peripheral atoms by expanding/contracting the square matrix.)

0 = \begin{vmatrix}

\cos \theta_{11} & \cos \theta_{12} & \cos \theta_{13} & \cos \theta_{14} \\

\cos \theta_{21} & \cos \theta_{22} & \cos \theta_{23} & \cos \theta_{24} \\

\cos \theta_{31} & \cos \theta_{32} & \cos \theta_{33} & \cos \theta_{34} \\

\cos \theta_{41} & \cos \theta_{42} & \cos \theta_{43} & \cos \theta_{44} \end{vmatrix}

Molecular geometry is determined by the quantum mechanical behavior of the electrons. Using the valence bond approximation this can be understood by the type of bonds between the atoms that make up the molecule. When atoms interact to form a chemical bond, the atomic orbitals of each atom are said to combine in a process called orbital hybridisation. The two most common types of bonds are sigma bonds (usually formed by hybrid orbitals) and pi bonds (formed by unhybridized p orbitals for atoms of main group elements). The geometry can also be understood by molecular orbital theory where the electrons are delocalised.

An understanding of the wavelike behavior of electrons in atoms and molecules is the subject of quantum chemistry

QMRIn graph theory, a star Sk is the complete bipartite graph K1,k: a tree with one internal node and k leaves (but, no internal nodes and k + 1 leaves when k ≤ 1). Alternatively, some authors define Sk to be the tree of order k with maximum diameter 2; in which case a star of k > 2 has k − 1 leaves.

S4 is a quadrant

QMR The fourth square is always different.Sprouts (game)

From Wikipedia, the free encyclopedia

"Brussels Sprouts (game)" redirects here. For the vegetable, see Brussels sprout.

A 2-spot game of Sprouts. The game ends when the first player is unable to draw a connecting line between the only two free points, marked in green.

Sprouts is a pencil-and-paper game with significant mathematical properties. It was invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in the early 1960s

Rules[edit]

The game is played by two players, starting with a few spots drawn on a sheet of paper. Players take turns, where each turn consists of drawing a line between two spots (or from a spot to itself) and adding a new spot somewhere along the line. The players are constrained by the following rules.

The line may be straight or curved, but must not touch or cross itself or any other line.

The new spot cannot be placed on top of one of the endpoints of the new line. Thus the new spot splits the line into two shorter lines.

No spot may have more than three lines attached to it. For the purposes of this rule, a line from the spot to itself counts as two attached lines and new spots are counted as having two lines already attached to them.

In so-called normal play, the player who makes the last move wins. In misère play, the player who makes the last move loses. (Misère Sprouts is perhaps the only misère combinatorial game that is played competitively in an organized forum.[1])

The diagram on the right shows a 2-spot game of normal-play Sprouts. After the fourth move, most of the spots are dead–they have three lines attached to them, so they cannot be used as endpoints for a new line. There are two spots (shown in green) that are still alive, having fewer than three lines attached. However, it is impossible to make another move, because a line from a live spot to itself would make four attachments, and a line from one live spot to the other would cross lines. Therefore, no fifth move is possible, and the first player loses. Live spots at the end of the game are called survivors and play a key role in the analysis of Sprouts.

A variant of the game, called Brussels Sprouts after the vegetable, starts with a number of crosses, i.e. spots with four free ends. Each move involves joining two free ends with a curve (again not crossing any existing line) and then putting a short stroke across the line to create two new free ends. This game is finite, and the total number of moves is predetermined by the initial number of crosses: the players cannot affect the result by their play.

Each move removes two free ends and introduces two more. With n initial crosses, the number of moves will be 5n−2, so a game starting with an odd number of crosses will be a first player win, while a game starting with an even number will be a second player win regardless of the moves.

To prove this (assuming that the game ends), let m denote the number of moves and v,e,f denote the number of vertices, edges, and faces of the planar graph obtained at the end of the game, respectively. We have:

e = 2m since at each move, the player adds 2 edges.

v = n + m since at each move, the player adds one vertex and the game starts with n vertices.

f = 4n since there is exactly one free end in each face at the end of the game, and the number of free ends does not change during the game.

The Euler characteristic for planar graphs is 2, so 2 = f-e+v = 4n-2m+n+m = 5n-m, hence m = 5n-2.

QMROne of the earliest known mathematicians was Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.[1] He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem.

QMRTetractys

Pythagoras was also credited with devising the tetractys, the triangular figure of four rows which add up to the perfect number, ten. As a mystical symbol, it was very important to the worship of the Pythagoreans who would swear oaths by it.

Iamblichus, Vit. Pyth., 29

"And the inventions were so admirable, and so divinised by those who understood them, that the members used them as forms of oath: 'By him who handed to our generation the tetractys, source of the roots of ever-flowing nature.'"

Brewer (1894), wrote (page 2732):[26]

"The four letters, meaning the four which compose the name of Deity. The ancient Jews never pronounced the word Jehovah composed of the four sacred letters JHVH. The word means 'I am,' or 'I exist' (Exod. iii. 14); but Rabbi Bechai says the letters include the three times— past, present, and future. Pythagoras called Deity a Tetrad or Tetractys, meaning the 'four sacred letters.'"

Religion and science

Heraclides Ponticus reports the story that Pythagoras claimed that he had lived four previous lives that he could remember in detail.[69] One of his past lives, as reported by Aulus Gellius, was as a beautiful courtesan.[70] According to Xenophanes, Pythagoras heard the cry of his dead friend in the bark of a dog.[71]

Brewer (1894), wrote (page 2293):[26]

"Pythagoras maintained that the soul has three vehicles: (1) the ethereal, which is luminous and celestial, in which the soul resides in a state of bliss in the stars; (2) the luminous, which suffers the punishment of sin after death; and (3) the terrestrial, which is the vehicle it occupies on this earth."

"Pythagoras asserted he could write on the moon. His plan of operation was to write on a looking—glass in blood, and place it opposite the moon, when the inscription would appear photographed or reflected on the moon's disc."

"Mesmerism was practised by Pythagoras, if we may credit Iamblichus, who tells us that he tamed a savage Daunian bear by “stroking it gently with his hand;” subdued an eagle by the same means; and held absolute dominion over beasts and birds by 'the power of his voice,' or 'influence of his touch.'"

"Pythagoras taught that the sun is a movable sphere in the centre of the universe, and that all the planets revolve round it. This is substantially the same as the Copernican and Newtonian systems."

"The Pythian games were held by the Greeks at Pytho, in Phocis, subsequently called Delphi. They took place every fourth year, the second of each Olympiad."

Quadrant Model of Reality

Monday, February 22, 2016

Quadrant Model of Reality Book 10 Philosophy

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