Quadrant Model of Reality Book 22 Philosophy

Philosophy Chapter

QMRThe first pupils of Aristotle commentated on his writings, but often with a view to expand his work. Thus Theophrastus invented five moods of syllogism in the first figure, in addition to the four invented by Aristotle, and stated with additional accuracy the rules of hypothetical syllogisms. He also often differed with his master,[1] including in collecting much information concerning animals and natural events, which Aristotle had omitted.

QMREthical categories[edit]

Another possible system for categorizing different schools of thought on war can be found in the Stanford Encyclopedia of Philosophy (see external links, below), based on ethics. The SEP describes three major divisions in the ethics of war: the Realist, the Pacifist, and the Just War Theory. In a nutshell:

Realists will typically hold that systems of morals and ethics which guide individuals within societies cannot realistically be applied to societies as a whole to govern the way they, as societies, interact with other societies. Hence, a state's purposes in war is simply to preserve its national interest. This kind of thinking is similar to Machiavelli's philosophy, and Thucydides and Hobbes may also fall under this category.

Pacifism however, maintains that a moral evaluation of war is possible, and that war is always found to be immoral. Generally, there are two kinds of modern secular pacifism to consider: (1) a more consequentialist form of pacifism (or CP), which maintains that the benefits accruing from war can never outweigh the costs of fighting it; and (2) a more deontological form of pacifism (or DP), which contends that the very activity of war is intrinsically wrong, since it violates foremost duties of justice, such as not killing human beings. Henry Ford and others were famous advocates of pacifistic diplomatic methods instead of war.

Nonviolence also holds that a moral evaluation of war is a duty, and that war is always found to be immoral. Mohandas K. Gandhi, Martin Luther King and Leo Tolstoy were all famous advocates of power of truth, lawfulness, soft power, nonviolent resistance and civil disobedience methods instead of war and to prevent war. Gandhi said he disliked more cowardice than violence.

Just War Theory, along with pacifism, holds that morals do apply to war. However, unlike pacifism, according to Just War Theory it is possible for a war to be morally justified. The concept of a morally justified war underlies much of the concept International Law, such as the Geneva Conventions. Aristotle, Cicero, Augustine, Aquinas, and Hugo Grotius are among the philosophers who have espoused some form of a just war philosophy. One common Just War Theory evaluation of war is that war is only justified if 1.) waged

in a state or nation's self-defense, or 2.) waged in order to end gross violations of human rights. Political philosopher John Rawls advocated these criteria as justification for war.

QMR Indian war elephants carried four men. Two shot from the sides and one shot from behind and one guided the elephant

QMR. One reason hypothesized why the US won WWII was Eisenhower had only four commanders under him. One for the navy air force the US army group and the British army group, whereas Hitler had a very complicated structure under him. The US had a very clean orb chart with well defined responsibilities with the four commanders under Eisenhower, whereas the the Germans had a very complicated structure that did not work well.

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."

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.

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.

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.

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

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

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 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

Construction by alternation[edit]

Vertex figure with nonplanar 3.3.3.3 vertex configuration for the triangular bipyramids

The geometry can also be constructed with an alternation operation applied to a hexagonal prismatic honeycomb. The hexagonal prism cells become octahedra and the voids create triangular bipyramids which can be divided into pairs of tetrahedra of this honeycomb. This honeycomb with bipyramids is called a ditetrahedral-octahedral honeycomb. There are 3 Coxeter-Dynkin diagrams, which can be seen as 1, 2, or 3 colors of octahedra:

CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node.png

CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node.png

CDel branch hh.pngCDel split2.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node.png

Gyroelongated alternated cubic honeycomb[edit]

Gyroelongated alternated cubic honeycomb

Type Uniform honeycomb

Schläfli symbol h{4,3,4}:ge

{3,6}h1{∞}

Coxeter diagram CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node 1.png

CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node 1.png

CDel branch hh.pngCDel split2.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node 1.png

Cell types {3,3}, {3,4}, (3.4.4)

Face types {3}, {4}

Vertex figure Gyroelongated alternated cubic honeycomb verf.png

Space group P63/mmc (194)

[3,6,2+,∞]

Properties vertex-uniform

The gyroelongated alternated cubic honeycomb or elongated triangular antiprismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of octahedra, triangular prisms, and tetrahedra in a ratio of 1:2:2.

It is vertex-uniform with 3 octahedra, 4 tetrahedra, 6 triangular prisms around each vertex.

It is one of 28 convex uniform honeycombs.

The elongated alternated cubic honeycomb has the same arrangement of cells at each vertex, but the overall arrangement differs. In the elongated form, each prism meets a tetrahedron at one of its triangular faces and an octahedron at the other; in the gyroelongated form, the prism meets the same kind of deltahedron at each end.

Gyroelongated alternated cubic tiling.png Gyroelongated alternated cubic honeycomb.png

Elongated alternated cubic honeycomb[edit]

Elongated alternated cubic honeycomb

Type Uniform honeycomb

Schläfli symbol h{4,3,4}:e

{3,6}g1{∞}

Cell types {3,3}, {3,4}, (3.4.4)

Vertex figure Gyrated triangular prismatic honeycomb verf.png

triangular cupola joined to isosceles hexagonal pyramid

Space group ?

Properties vertex-transitive

The elongated alternated cubic honeycomb or elongated triangular gyroprismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of octahedra, triangular prisms, and tetrahedra in a ratio of 1:2:2. It is vertex-uniform with 3 octahedra, 4 tetrahedra, 6 triangular prisms around each vertex. Each prism meets an octahedron at one end and a tetrahedron at the other. It is one of 28 convex uniform honeycombs. It has a gyrated form called the gyroelongated alternated cubic honeycomb with the same arrangement of cells at each vertex.

Gyrated tetrahedral-octahedral honeycomb[edit]

Gyrated tetrahedral-octahedral honeycomb

Type convex uniform honeycomb

Coxeter diagram CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node.png

CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node.png

CDel branch hh.pngCDel split2.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node.png

Schläfli symbol h{4,3,4}:g

h{6,3}h{∞}

s{3,6}h{∞}

s{3[3]}h{∞}

Cell types {3,3}, {3,4}

Vertex figure Gyrated alternated cubic honeycomb verf.png

Triangular orthobicupola G3.4.3.4

Space group P63/mmc (194)

[3,6,2+,∞]

Dual trapezo-rhombic dodecahedral honeycomb

Properties vertex-transitive, face-transitive

The gyrated tetrahedral-octahedral honeycomb or gyrated alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of octahedra and tetrahedra in a ratio of 1:2.

It is vertex-uniform with 8 tetrahedra and 6 octahedra around each vertex.

It is not edge-uniform. All edges have 2 tetrahedra and 2 octahedra, but some are alternating, and some are paired.

Gyrated alternated cubic.pngGyrated alternated cubic honeycomb.png

It can be seen as reflective layers of this layer honeycomb:

CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png

Tetroctahedric semicheck.png

Construction by gyration[edit]

This is a less symmetric version of another honeycomb, tetrahedral-octahedral honeycomb, in which each edge is surrounded by alternating tetrahedra and octahedra. Both can be considered as consisting of layers one cell thick, within which the two kinds of cell strictly alternate. Because the faces on the planes separating these layers form a regular pattern of triangles, adjacent layers can be placed so that each octahedron in one layer meets a tetrahedron in the next layer, or so that each cell meets a cell of its own kind (the layer boundary thus becomes a reflection plane). The latter form is called gyrated.

The vertex figure is called a triangular orthobicupola, compared to the tetrahedral-octahedral honeycomb whose vertex figure cuboctahedron in a lower symmetry is called a triangular gyrobicupola, so the gyro- prefix is reversed in usage.

QMRThe runcic cubic honeycomb or runcicantic cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of rhombicuboctahedra, cubes, and tetrahedra in a ratio of 1:1:2. Its vertex figure is a triangular prism, with a tetrahedron on one end, cube on the opposite end, and three rhombicuboctahedra around the trapezoidal sides.

John Horton Conway calls this honeycomb a 3-RCO-trille, and its dual quarter cubille.

Runcinated alternated cubic tiling.pngHC A5-P2-P1.png

Related honeycombs[edit]

It is related to the runcinated cubic honycomb, with quarter of the cubes alternated into tetrahedra, and half expanded into rhombicuboctahedra.

Runcinated cubic honeycomb.png

Runcinated cubic

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png Runcinated alternated cubic honeycomb.jpg

Runcic cubic

CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node 1.png

{4,3}, {4,3}, {4,3}, {4,3}

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png, CDel node 1.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node 1.png, CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png h{4,3}, rr{4,3}, {4,3} CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png, CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png This honeycomb can be divided on truncated square tiling planes, using the octagons centers of the rhombicuboctahedra, creating square cupolae. This scaliform honeycomb is represented by Coxeter diagram CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png, and symbol s3{2,4,4}, with coxeter notation symmetry [2+,4,4].

QMRCantic cubic honeycomb[edit]

Cantic cubic honeycomb

Type Uniform honeycomb

Schläfli symbol h2{4,3,4}

Coxeter diagrams CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png

CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png

Cells t{3,4} Uniform polyhedron-43-t12.png

r{4,3} Uniform polyhedron-43-t1.png

t{3,3} Uniform polyhedron-33-t01.png

Vertex figure Truncated alternated cubic honeycomb verf.png

Coxeter groups [4,31,1], {\tilde{B}}_3

[3[4]], {\tilde{A}}_3

Symmetry group Fm3m (225)

Dual half oblate octahedrille

Properties vertex-transitive

The cantic cubic honeycomb, cantic cubic cellulation or truncated half cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated octahedra, cuboctahedra and truncated tetrahedra in a ratio of 1:1:2. Its vertex figure is a rectangular pyramid.

John Horton Conway calls this honeycomb a truncated tetraoctahedrille, and its dual half oblate octahedrille. Truncated alternated cubic tiling.png HC A1-A3-A4.png Symmetry[edit] It has two different uniform constructions. The {\tilde{A}}_3 construction can be seen with alternately colored truncated tetrahedra.

Its vertex arrangement represents an A3 lattice or D3 lattice.[2][3] It is the 3-dimensional case of a simplectic honeycomb. Its Voronoi cell is a rhombic dodecahedron, the dual of the cuboctahedron vertex figure for the tet-oct honeycomb.

The D+

3 packing can be constructed by the union of two D3 (or A3) lattices. The D+

n packing is only a lattice for even dimensions. The kissing number is 22=4, (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[4]

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png ∪ CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node 1.png

The A*

3 or D*

3 lattice (also called A4

3 or D4

3) can be constructed by the union of all four A3 lattices, and is identical to the vertex arrangement of the disphenoid tetrahedral honeycomb, dual honeycomb of the uniform bitruncated cubic honeycomb:[5] It is also the body centered cubic, the union of two cubic honeycombs in dual positions. CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png ∪ CDel node.pngCDel split1.pngCDel nodes 10luru.pngCDel split2.pngCDel node.png ∪ CDel node.pngCDel split1.pngCDel nodes 01lr.pngCDel split2.pngCDel node.png ∪ CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node 1.png = dual of CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png ∪ CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png. The kissing number of the D* 3 lattice is 8[6] and its Voronoi tessellation is a bitruncated cubic honeycomb, CDel branch 11.pngCDel 4a4b.pngCDel nodes.png, containing all truncated octahedral Voronoi cells, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png.[7]

Alternated cubic honeycomb slices[edit] The alternated cubic honeycomb can be sliced into sections, where new square faces are created from inside of the octahedron. Each slice will contain up and downward facing square pyramids and tetrahedra sitting on their edges. A second slice direction needs no new faces and includes alternating tetrahedral and octahedral. This slab honeycomb is a scaliform honeycomb rather than uniform because it has nonuniform cells.

QMRThe tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating octahedra and tetrahedra in a ratio of 1:2.

Other names include half cubic honeycomb, half cubic cellulation, or tetragonal disphenoidal cellulation. John Horton Conway calls this honeycomb a tetroctahedrille, and its dual dodecahedrille.

It is vertex-transitive with 8 tetrahedra and 6 octahedra around each vertex. It is edge-transitive with 2 tetrahedra and 2 octahedra alternating on each edge.

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

It is part of an infinite family of uniform honeycombs called alternated hypercubic honeycombs, formed as an alternation of a hypercubic honeycomb and being composed of demihypercube and cross-polytope facets. It is also part of another infinite family of uniform honeycombs called simplectic honeycombs. In this case of 3-space, the cubic honeycomb is alternated, reducing the cubic cells to tetrahedra, and the deleted vertices create octahedral voids. As such it can be represented by an extended Schläfli symbol h{4,3,4} as containing half the vertices of the {4,3,4} cubic honeycomb. There's a similar honeycomb called gyrated tetrahedral-octahedral honeycomb which has layers rotated 60 degrees so half the edges have neighboring rather than alternating tetrahedra and octahedra.

Barrel vaults. This type of vault has a cross section of a simple arch. Usually this type of space frame does not need to use tetrahedral modules or pyramids as a part of its backing. Spherical domes and other compound curves usually require the use of tetrahedral modules or pyramids and additional support from a skin.

Overview[edit] Simplified space frame roof with the half-octahedron highlighted in blue The simplest form of space frame is a horizontal slab of interlocking square pyramids and tetrahedra built from aluminium or tubular steel struts. In many ways this looks like the horizontal jib of a tower crane repeated many times to make it wider. A stronger form is composed of interlocking tetrahedra in which all the struts have unit length. More technically this is referred to as an isotropic vector matrix or in a single unit width an octet truss. More complex variations change the lengths of the struts to curve the overall structure or may incorporate other geometrical shapes.

QMRSpace frames were independently developed by Alexander Graham Bell around 1900 and Buckminster Fuller in the 1950s. Bell's interest was primarily in using them to make rigid frames for nautical and aeronautical engineering, with the tetrahedral truss being one of his inventions. However few of his designs were realised. Buckminster Fuller's focus was architectural structures; his work had greater influence. Introduction of the first space grid system called MERO in 1943 in Germany initiated the use of space trusses in architecture. The commonly used method, still in use has individual tubular members connected at node joints (ball shaped). Different systems like space deck system, octet truss system, Pyramitec system, Unibat system, Cubic system, etc. were developed. A method of Tree supports was developed to replace the individual columns.[1]

QMRHow I (currently) Explain The State of Blockchains To Executives and Researchers

Seems everyone these days is getting into the game of providing frameworks for thinking about blockchains and trying to convince others that their definitions are the correct definitions. Into that marketplace of metaphors, I provide this entry.

When I look at the development of the sector generally, what I see is roughly two “types” of blockchains and within each of those types, two “flavours” of blockchains. These types and flavours can be graphically depicted using two dimensions.

On the X axis of the diagram we can formulate a spectrum of permission-able-ness. These permissions are usually capabilities based permissions, meaning the permission is to interact with a capability of what the blockchain can do. Whether a blockchain design is capable of being put into permissioned mode or not is an important operational consideration for application developers (whether those are startups or enterprises). That permission layer may provide an advantage in censorship resistance (if it is absent) or in compliance risk mitigation (if it is present).

On the Y axis of the diagram we can formulate a spectrum of optimizations. These optimizations are roughly binary at this point, although we fully expect that it will be more of a spectrum that will develop over time. On one side we have transactional optimized blockchains. These are the chain designs which have been developed to support digital cash and are now being permissioned and built to provide clearing and settlement solutions. On the other side we have logic optimized blockchains. These blockchains have been optimized to provide an arbitrary framework for running small scripts which are saved onto the blockchain (which we call “smart contracts”).

In total, my mental diagram looks like this:

While the above may not perfectly capture all of the blockchains in existence, I think it does a fairly good job of providing a framework for placing most of the space into some easier to consume boxes.

The Optimization Spectrum

On the lower half of the quadrant we have blockchains which give application developers a clear and efficient way to verifiably track title transfers in a distributed environment. Whether these blockchains are permissioned or unpermissioned, they are a good fit for application developers seeking to build transfer mechanisms, clearing and settlement, and provenance applications. In other words, they’re really interesting property registers. These blockchain designs – to some extent or another depending on the blockchain in question – do provide some limited logic capabilities (bitcoin, famously, has its multi-signature capacity which operates in a similar area to logic). However, they really have been optimized to track movement of title over “property” from one node on the network to another.

On the upper half of the quadrant we have blockchains which give application developers a clear and efficient way to verifiably track business and governance process logic in a distributed environment. Whether these blockchains are permissioned or unpermissioned, they are a good fit for application developers seeking to build complicated business process mechanisms. In other words, they’re really interesting process auditors. Similarly to transaction optimized blockchain designs, they have capabilities of supporting verifiable title transfers, but they have really been optimized to run arbitrary business logic.

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.

The Permissioned Spectrum

On the left half of the quandrant we have unpermissioned blockchains. These blockchains lack an access control layer and as such handle anti-spam and consensus via purely economic mechanisms. We may not like to have to pay a bank a fee to update our address in their database, but if our bank operates on a public blockchain that’s basically what we’d have to do in order to overcome the necessary anti-spam protections (and other protections) which have been put in place to protect these unpermissioned blockchains. These blockchains are the best solution for censorship resistance. If someone needs data to exist forever in a rock solid vault of math and environmental degradation, then public blockchains are the place for that data. Public blockchains also have public governance mechanisms, as we are finding out with the blockchain debate. Whether the increased uncertainty which is the current state of the public blockchain governance oligarchy is a good or a bad thing remains to be seen. Lastly, public blockchains have been designed to provide the backbone for a variety of applications. That means that they were probably not well suited for any one type of application. Depending on what application one is seeking to build this may be a benefit or a detriment.

On the right half of the quandrant we have blockchains which are capable of being put into a permissioned mode. Generally speaking, permissions can be made fully public, or use whitelisting to control who can validate batches of transactions, who can add functionality to the blockchain in the form of smart contracts, and who can transact with the chain. These chains are not susceptible to external attack by unknown actors because the clients running the chain will reject blocks from not-whitelisted nodes (if the client is running in “permissioned” mode for a particular blockchain in question). These chains also may have slight performance advantages over public blockchains because they are only dealing with the functionality required for that chain rather than all the functionality for all of the people for all of the time. While permissioned chains have some upsides, they also have some downsides of course. The downsides include a reduction in censorship resistance, and an increase in responsibility for application developers (who now have to also have some operational responsibilities). Hope that helps your own mental framing of the state of blockchain technology.

QMREuclid's fifth postulate implies Playfair's postulate[edit] The easiest way to show this is using the Euclidean theorem (equivalent to the fifth postulate) that states that the angles of a triangle sum to two right angles. Given a line \ell and a point P not on that line, construct a line, t, perpendicular to the given one through the point P, and then a perpendicular to this perpendicular at the point P. This line is parallel because it cannot meet \ell and form a triangle.[12] Now it can be seen that no other parallels exist. If n was a second line through P, then n makes an acute angle with t (since it is not the perpendicular) and the hypothesis of the fifth postulate holds, and so, n meets \ell.[13] Playfair's implies Euclid's fifth postulate[edit] Given that Playfair's postulate implies that only the perpendicular to the perpendicular is a parallel, the lines of the Euclid construction will have to cut each other in a point. It is also necessary to prove that they will do it in the side where the angles sum to less than two right angles, but this is more difficult.[14]

History Chapter

QMRIn 632 Qi helped Jin defeat Chu at the Battle of Chengpu. In 589 Qi was defeated by Jin. In 579 the four great powers of Qin (west), Jin (center), Chu (south) and Qi (east) met to declare a truce and limit their military strength. In 546 a similar four-power conference recognized several smaller states as satellites of Qi, Jin and Qin.

At the end of the Kamakura period (1333) the four temples of Kennin-ji, Kenchō-ji, Engaku-ji and Jufuku-ji, were already known as the Gozan, but not much is otherwise known about the system, its structure and the hierarchical order.[1] The first official recognition of the system came from Emperor Go-Daigo during the brief Kemmu Restoration (1333–1336). Go-Daigo added the Kyoto Gozan to the existing temples in Kamakura with Daitoku-ji and Nanzen-ji together at the top as number 1, followed by Kennin-ji and Tōfuku-ji. At this point in time, in spite of their name, the Gozan were not five but four in both cities.[1] At the beginning of Muromachi Period, they became five in Kyoto later, when Ashikaga Takauji built Tenryū-ji in memory of Go-Daigo.

Source for the Arthurian legend[edit] There are two entries in the Annales on King Arthur, one on Medraut (Mordred), and one on Merlin. These entries have been presented in the past as proof of the existence of Arthur and Merlin,[2] although that view is no longer widely held because the entries could have been added arbitrarily as late as 970, long after the development of the early Arthurian myth. The entries on Arthur, Mordred, and Merlin in the A Text: Year 72 (c. AD 516) The Battle of Badon, in which Arthur carried the cross of our Lord Jesus Christ on his shoulders for three days and three nights and the Britons were victors. Year 93 (c. 537) The Strife of Camlann in which Arthur and Mordred fell and there was death in Britain and in Ireland. Year 129 (c. 573) The Battle of Armterid Texts B and C omit the second half of the year 93 entry. B calls Arfderydd "Erderit"; C, "Arderit". In the B Text, the year 129 entry continues: "between the sons of Elifer and Guendoleu son of Keidau in which battle Guendoleu fell and Merlin went mad". Concerning Arthur's cross at the Battle of Badon, it is mirrored by a passage in Nennius where Arthur was said to have borne the image of the Virgin Mary "on his shoulders" during a battle at a castle called Guinnion.[3] The words for "shoulder" and "shield" were, however, easily confused in Old Welsh – *scuit "shield" versus *scuid "shoulder" [3] – and Geoffrey of Monmouth played upon this dual tradition, describing Arthur bearing "on his shoulders a shield" emblazoned with the Virgin.[4]

QMRThe principal versions of Annales Cambriae appear in four manuscripts:

A: London, British Library, MS. Harleian 3859, folios 190r-193r.

B: London (Kew), Public Record Office, MS. E.164/1 (K.R. Misc. Books, Series I) pp. 2–26

C: London, British Library, MS. Cotton Domitian A.i, folios 138r-155r

D: Exeter, Cathedral Library, MS. 3514, pp. 523–28, the Cronica ante aduentum Domini.

E: ibid., pp. 507–19, the Cronica de Wallia.

A is written in a hand of about 1100x1130 AD, and inserted without title into a manuscript (MS) of the Historia Brittonum where it is immediately followed by a pedigree for Owain ap Hywel (died 988). Although no explicit chronology is given in the MS, its annals seem to run from about AD 445 to 977 with the last entry at 954, making it likely that the text belongs to the second half of the 10th century.

B was written, probably at the Cistercian abbey of Neath, at the end of the 13th century. It is entitled Annales ab orbe condito adusque A. D. mcclxxxvi [1286].

C is part of a book written at St David's, and is entitled Annales ab orbe condito adusque A. D. mcclxxviii [1288][or 1278?]; this is also of the late 13th century. Two of the texts, B and C, begin with a World Chronicle derived from Isidore of Seville's Origines (Book V, ch. 39), through the medium of Bede's Chronica minora. B commences its annals with Julius Caesar's invasion of Britain "sixty years before the incarnation of the Lord." After A.D. 457, B agrees closely with A until A ends. C commences its annals after the empire of Heraclius (AD 610-41) at a year corresponding to AD 677. C mostly agrees with A until A ends, although it is clear that A was not the common source for B and C (Dumville 2002, p. xi). B and C diverge after 1203, C having fewer and briefer Welsh entries. D and E are found in a manuscript written at the Cistercian abbey of Whitland in south-west Wales in the later 13th century; the Cronica ante aduentum Domini (which takes its title from its opening words) extends from 1132 BC to 1285 AD, while the Cronica de Wallia extends from 1190 to 1266. A alone has benefited from a complete diplomatic edition (Phillimore 1888).[1]

QMRHistorical method basics

The following questions are used by historians in modern work.

When was the source, written or unwritten, produced (date)?

Where was it produced (localization)?

By whom was it produced (authorship)?

From what pre-existing material was it produced (analysis)?

In what original form was it produced (integrity)?

What is the evidential value of its contents (credibility)?

The first four are known as higher criticism; the fifth, lower criticism; and, together, external criticism. The sixth and final inquiry about a source is called internal criticism.