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65-603: The origin of the Sonobe module is unknown. Two possible creators are Toshie Takahama and Mitsunobu Sonobe, who published several books together and were both members of Sōsaku Origami Gurūpu '67. The earliest appearance of a Sonobe module was in a cube attributed to Mitsunobu Sonobe in the Sōsaku Origami Gurūpu '67's magazine Origami in Issue 2 (1968). It does not reveal whether he invented the module or used an earlier design;

130-400: A 1 a 2 … a n b 1 b 2 … b n ] ∈ R 2 × n {\displaystyle V={\begin{bmatrix}a_{1}&a_{2}&\dots &a_{n}\\b_{1}&b_{2}&\dots &b_{n}\end{bmatrix}}\in \mathbb {R} ^{2\times n}} . Then

195-552: A logical system in which each result is proved from axioms and previously proved theorems. The Elements begins with plane geometry , still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs . It goes on to the solid geometry of three dimensions . Much of the Elements states results of what are now called algebra and number theory , explained in geometrical language. For more than two thousand years,

260-415: A rectangle with a width of 3 and a length of 4 has an area that represents the product, 12. Because this geometrical interpretation of multiplication was limited to three dimensions, there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are implied, for example in the proof of book IX, proposition 20. Euclid refers to a pair of lines, or

325-407: A trapezoid and a right triangle , and rearranged into a rectangle , as shown in the figure to the left. This means that the area of a parallelogram is the same as that of a rectangle with the same base and height: The base × height area formula can also be derived using the figure to the right. The area K of the parallelogram to the right (the blue area) is the total area of the rectangle less

390-402: A 2x6 rectangle and a 3x4 rectangle are equal but not congruent, and the letter R is congruent to its mirror image. Figures that would be congruent except for their differing sizes are referred to as similar . Corresponding angles in a pair of similar shapes are equal and corresponding sides are in proportion to each other. Because of Euclidean geometry's fundamental status in mathematics, it

455-442: A bounding parallelogram, formed by the tangent lines to the ellipse at the four endpoints of the conjugate diameters. All tangent parallelograms for a given ellipse have the same area. It is possible to reconstruct an ellipse from any pair of conjugate diameters, or from any tangent parallelogram. A parallelepiped is a three-dimensional figure whose six faces are parallelograms. Euclidean geometry Euclidean geometry

520-577: A colorful figure about whom many historical anecdotes are recorded, is remembered along with Euclid as one of the greatest of ancient mathematicians. Although the foundations of his work were put in place by Euclid, his work, unlike Euclid's, is believed to have been entirely original. He proved equations for the volumes and areas of various figures in two and three dimensions, and enunciated the Archimedean property of finite numbers. Apollonius of Perga ( c.  240 BCE  – c.  190 BCE )

585-427: A pair of planar or solid figures, as "equal" (ἴσος) if their lengths, areas, or volumes are equal respectively, and similarly for angles. The stronger term " congruent " refers to the idea that an entire figure is the same size and shape as another figure. Alternatively, two figures are congruent if one can be moved on top of the other so that it matches up with it exactly. (Flipping it over is allowed.) Thus, for example,

650-437: A parallelogram are equal in length. Therefore, triangles ABE and CDE are congruent (ASA postulate, two corresponding angles and the included side ). Therefore, Since the diagonals AC and BD divide each other into segments of equal length, the diagonals bisect each other. Separately, since the diagonals AC and BD bisect each other at point E , point E is the midpoint of each diagonal. Parallelograms can tile

715-477: A parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. By comparison, a quadrilateral with at least one pair of parallel sides

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780-404: A ray as an object that extends to infinity in one direction, would normally use locutions such as "if the line is extended to a sufficient length", although he occasionally referred to "infinite lines". A "line" for Euclid could be either straight or curved, and he used the more specific term "straight line" when necessary. The pons asinorum ( bridge of asses ) states that in isosceles triangles

845-472: A regular cubic grid, which can be easily derived from the six unit cube by joining multiple ones at faces or edges. There are two popular variants of the main assembly style of three modules in triangular pyramids, both using the same flaps and pockets and compatible with it: Parallelogram In Euclidean geometry , a parallelogram is a simple (non- self-intersecting ) quadrilateral with two pairs of parallel sides. The opposite or facing sides of

910-451: A right-angle pyramid consisting of one half each of three units, without dangling flaps. The pyramids can be made to point inwards; assembly is more difficult but some cases of encroaching can be obviously prevented. The simplest shape made of these pyramids, often called "Toshie's Jewel" (shown above), is named after origami artist Toshie Takahama , who first printed a diagram of this in her 1974 book Creative Life with Creative Origami . It

975-445: A statement such as "Find the greatest common measure of ..." Euclid often used proof by contradiction . Points are customarily named using capital letters of the alphabet. Other figures, such as lines, triangles, or circles, are named by listing a sufficient number of points to pick them out unambiguously from the relevant figure, e.g., triangle ABC would typically be a triangle with vertices at points A, B, and C. Angles whose sum

1040-447: A steep bridge that only a sure-footed donkey could cross. Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26). Triangles with three equal angles (AAA) are similar, but not necessarily congruent. Also, triangles with two equal sides and an adjacent angle are not necessarily equal or congruent. The sum of

1105-453: Is constructive . Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge . In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory , which often assert

1170-587: Is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped . The word "parallelogram" comes from the Greek παραλληλό-γραμμον, parallēló-grammon , which means "a shape of parallel lines". A simple (non-self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements

1235-421: Is a mathematical system attributed to ancient Greek mathematician Euclid , which he described in his textbook on geometry , Elements . Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions ( theorems ) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into

1300-403: Is a right angle are called complementary . Complementary angles are formed when a ray shares the same vertex and is pointed in a direction that is in between the two original rays that form the right angle. The number of rays in between the two original rays is infinite. Angles whose sum is a straight angle are supplementary . Supplementary angles are formed when a ray shares the same vertex and

1365-408: Is a three-unit hexahedron built around the notional scaffold of a flat equilateral triangle (two "faces", three edges); the protruding tab/pocket flaps are simply reconnected on the underside, resulting in two triangular pyramids joined at the base, a triangular bipyramid . A popular intermediate model is the triakis icosahedron , shown below. It requires 30 units to build. The table below shows

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1430-401: Is equivalent to the determinant of a matrix built using a , b and c as rows with the last column padded using ones as follows: To prove that the diagonals of a parallelogram bisect each other, we will use congruent triangles : (since these are angles that a transversal makes with parallel lines AB and DC ). Also, side AB is equal in length to side DC , since opposite sides of

1495-438: Is impractical to give more than a representative sampling of applications here. As suggested by the etymology of the word, one of the earliest reasons for interest in and also one of the most common current uses of geometry is surveying . In addition it has been used in classical mechanics and the cognitive and computational approaches to visual perception of objects . Certain practical results from Euclidean geometry (such as

1560-466: Is in contrast to analytic geometry , introduced almost 2,000 years later by René Descartes , which uses coordinates to express geometric properties by means of algebraic formulas . The Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was rapidly recognized, with the result that there was little interest in preserving the earlier ones, and they are now nearly all lost. There are 13 books in

1625-512: Is mainly known for his investigation of conic sections. René Descartes (1596–1650) developed analytic geometry , an alternative method for formalizing geometry which focused on turning geometry into algebra. In this approach, a point on a plane is represented by its Cartesian ( x , y ) coordinates, a line is represented by its equation, and so on. In Euclid's original approach, the Pythagorean theorem follows from Euclid's axioms. In

1690-444: Is pointed in a direction that is in between the two original rays that form the straight angle (180 degree angle). The number of rays in between the two original rays is infinite. In modern terminology, angles would normally be measured in degrees or radians . Modern school textbooks often define separate figures called lines (infinite), rays (semi-infinite), and line segments (of finite length). Euclid, rather than discussing

1755-485: Is proved that there are infinitely many prime numbers. Books XI–XIII concern solid geometry . A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. The platonic solids are constructed. Euclidean geometry is an axiomatic system , in which all theorems ("true statements") are derived from a small number of simple axioms. Until the advent of non-Euclidean geometry , these axioms were considered to be obviously true in

1820-403: Is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the gravitational field ). Euclidean geometry is an example of synthetic geometry , in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects. This

1885-499: Is the determination of packing arrangements , such as the problem of finding the most efficient packing of spheres in n dimensions. This problem has applications in error detection and correction . Geometry is used extensively in architecture . Geometry can be used to design origami . Some classical construction problems of geometry are impossible using compass and straightedge , but can be solved using origami . Archimedes ( c.  287 BCE  – c.  212 BCE ),

1950-507: Is the addition of secondary units to basic Sonobe unit forms to create new geometric shapes; some of which can be seen in Tomoko Fuse's book Unit Origami: Multidimensional Transformations (1990). Each individual unit is folded from a square sheet of paper, of which only one face is visible in the finished module; many ornamented variants of the plain Sonobe unit that expose both sides of the paper have been designed. The Sonobe unit has

2015-407: Is true: Thus, all parallelograms have all the properties listed above, and conversely , if just any one of these statements is true in a simple quadrilateral, then it is considered a parallelogram. All of the area formulas for general convex quadrilaterals apply to parallelograms. Further formulas are specific to parallelograms: A parallelogram with base b and height h can be divided into

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2080-660: The Elements : Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, for example, "In any triangle, two angles taken together in any manner are less than two right angles." (Book I proposition 17) and the Pythagorean theorem "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." (Book I, proposition 47) Books V and VII–X deal with number theory , with numbers treated geometrically as lengths of line segments or areas of surface regions. Notions such as prime numbers and rational and irrational numbers are introduced. It

2145-550: The right angle as his basic unit, so that, for example, a 45- degree angle would be referred to as half of a right angle. The distance scale is relative; one arbitrarily picks a line segment with a certain nonzero length as the unit, and other distances are expressed in relation to it. Addition of distances is represented by a construction in which one line segment is copied onto the end of another line segment to extend its length, and similarly for subtraction. Measurements of area and volume are derived from distances. For example,

2210-680: The Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems. The equation defining the distance between two points P = ( p x , p y ) and Q = ( q x , q y ) is then known as the Euclidean metric , and other metrics define non-Euclidean geometries . In terms of analytic geometry,

2275-480: The adjective "Euclidean" was unnecessary because Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate ) that theorems proved from them were deemed absolutely true, and thus no other sorts of geometry were possible. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein 's theory of general relativity

2340-406: The angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another . Its name may be attributed to its frequent role as the first real test in the Elements of the intelligence of the reader and as a bridge to the harder propositions that followed. It might also be so named because of the geometrical figure's resemblance to

2405-426: The angles of a triangle is equal to a straight angle (180 degrees). This causes an equilateral triangle to have three interior angles of 60 degrees. Also, it causes every triangle to have at least two acute angles and up to one obtuse or right angle . The celebrated Pythagorean theorem (book I, proposition 47) states that in any right triangle, the area of the square whose side is the hypotenuse (the side opposite

2470-407: The area of a circle and the volume of a parallelepipedal solid. Euclid determined some, but not all, of the relevant constants of proportionality. For instance, it was his successor Archimedes who proved that a sphere has 2/3 the volume of the circumscribing cylinder. Euclidean geometry has two fundamental types of measurements: angle and distance . The angle scale is absolute, and Euclid uses

2535-456: The area of the Varignon parallelogram is half the area of the quadrilateral. Proof without words (see figure): For an ellipse , two diameters are said to be conjugate if and only if the tangent line to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding tangent parallelogram , sometimes called

2600-408: The area of the parallelogram generated by a and b is equal to det ( V V T ) {\displaystyle {\sqrt {\det(VV^{\mathrm {T} })}}} . Let points a , b , c ∈ R 2 {\displaystyle a,b,c\in \mathbb {R} ^{2}} . Then the signed area of the parallelogram with vertices at a , b and c

2665-412: The area of the two orange triangles. The area of the rectangle is and the area of a single triangle is Therefore, the area of the parallelogram is Another area formula, for two sides B and C and angle θ, is Provided that the parallelogram is not a rhombus, the area can be expressed using sides B and C and angle γ {\displaystyle \gamma } at the intersection of

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2730-476: The correlation between three basic characteristics – faces, edges, and vertices – of polygons (composed of Toshie's Jewel sub-units) of varying size and the number of Sonobe units used: The model made of three units results in a triangular bipyramid . Building a pyramid on each face of a regular tetrahedron , using six units, results in a cube (the central fold of each module lays flat, creating square faces instead of isosceles right triangular faces, and changing

2795-404: The cube and squaring the circle . In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve equations whose order is an integral power of two, while doubling a cube requires the solution of a third-order equation. Euler discussed a generalization of Euclidean geometry called affine geometry , which retains

2860-399: The diagonals: When the parallelogram is specified from the lengths B and C of two adjacent sides together with the length D 1 of either diagonal, then the area can be found from Heron's formula . Specifically it is where S = ( B + C + D 1 ) / 2 {\displaystyle S=(B+C+D_{1})/2} and the leading factor 2 comes from

2925-730: The existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory. Strictly speaking, the lines on paper are models of the objects defined within the formal system, rather than instances of those objects. For example, a Euclidean straight line has no width, but any real drawn line will have. Though nearly all modern mathematicians consider nonconstructive proofs just as sound as constructive ones, they are often considered less elegant , intuitive, or practically useful. Euclid's constructive proofs often supplanted fallacious nonconstructive ones, e.g. some Pythagorean proofs that assumed all numbers are rational, usually requiring

2990-446: The existence of the constructed objects, in his reasoning he also implicitly assumes them to be unique. The Elements also include the following five "common notions": Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more extensive and complete sets of axioms. To the ancients, the parallel postulate seemed less obvious than

3055-481: The faces of a dodecahedron a 90-module ball can be obtained. The 270-module ball looks like a very complicated shape, but it is just an icosahedron with each triangular face divided into 9 small triangles. Each small triangle is made of 3 sonobe units. Arbitrary shapes, beyond symmetrical polyhedra, can also be constructed; a deltahedron with 2N faces and 3N edges requires 3N Sonobe modules. A popular class of arbitrary shapes consists of assemblies of equal size cubes in

3120-576: The fact that the chosen diagonal divides the parallelogram into two congruent triangles. Let vectors a , b ∈ R 2 {\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{2}} and let V = [ a 1 a 2 b 1 b 2 ] ∈ R 2 × 2 {\displaystyle V={\begin{bmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{bmatrix}}\in \mathbb {R} ^{2\times 2}} denote

3185-423: The fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments (so line segments continue to have

3250-514: The formula for the number of faces, edges, and vertices), or triakis tetrahedron . Building a pyramid on each face of a regular octahedron , using twelve Sonobe units, results in a triakis octahedron . Building a pyramid on each face of a regular icosahedron requires 30 units, and results in a triakis icosahedron . Uniform polyhedra can be adapted to Sonobe modules by replacing non-triangular faces with pyramids having equilateral faces; for example by adding pentagonal pyramids pointing inwards to

3315-439: The manner of Euclid Book III, Prop. 31. In modern terminology, the area of a plane figure is proportional to the square of any of its linear dimensions, A ∝ L 2 {\displaystyle A\propto L^{2}} , and the volume of a solid to the cube, V ∝ L 3 {\displaystyle V\propto L^{3}} . Euclid proved these results in various special cases such as

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3380-530: The matrix with elements of a and b . Then the area of the parallelogram generated by a and b is equal to | det ( V ) | = | a 1 b 2 − a 2 b 1 | {\displaystyle |\det(V)|=|a_{1}b_{2}-a_{2}b_{1}|\,} . Let vectors a , b ∈ R n {\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{n}} and let V = [

3445-430: The number of special cases is reduced. Geometers of the 18th century struggled to define the boundaries of the Euclidean system. Many tried in vain to prove the fifth postulate from the first four. By 1763, at least 28 different proofs had been published, but all were found incorrect. Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example,

3510-428: The open bottom, and isosceles right triangles as the other three faces. It will have a right-angle apex (equivalent to the corner of a cube ) and three tab/pocket flaps protruding from the base. This particularly suits polyhedra that have equilateral triangular faces: Sonobe modules can replace each notional edge of the original deltahedron by the central diagonal fold of one unit and each equilateral triangle with

3575-466: The others, as evidenced by the organization of the Elements : his first 28 propositions are those that can be proved without it. Many alternative axioms can be formulated which are logically equivalent to the parallel postulate (in the context of the other axioms). For example, Playfair's axiom states: The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists. Euclidean Geometry

3640-462: The others. They aspired to create a system of absolutely certain propositions, and to them, it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible since one can construct consistent systems of geometry (obeying the other axioms) in which the parallel postulate is true, and others in which it is false. Euclid himself seems to have considered it as being qualitatively different from

3705-765: The phrase "finished model by Mitsunobu Sonobe" is ambiguous. The diagram from Issue 2 reappears in 1970 in the group's book Atarashii origami nyūmon (新しい折り紙入門 ). Another 1970s appearance of the unit was in the model "Toshie's Jewel", from Toshie Takahama's book Creative Life With Creative Origami Vol 1 . In the mid-1970s, Steve Krimball created the 30-unit ball, as mentioned in The Origamian vol. 13, no. 3 June 1976. Since then, many variations of modified Sonobe units have been developed; some examples of these can be found in Meenakshi Mukerji's book Marvelous Modular Origami (2007). Another variation to Sonobe models

3770-410: The physical world, so that all the theorems would be equally true. However, Euclid's reasoning from assumptions to conclusions remains valid independently from the physical reality. Near the beginning of the first book of the Elements , Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath): Although Euclid explicitly only asserts

3835-403: The plane by translation. If edges are equal, or angles are right, the symmetry of the lattice is higher. These represent the four Bravais lattices in 2 dimensions . An automedian triangle is one whose medians are in the same proportions as its sides (though in a different order). If ABC is an automedian triangle in which vertex A stands opposite the side a , G is the centroid (where

3900-437: The problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until Pierre Wantzel published a proof in 1837 that such a construction was impossible. Other constructions that were proved impossible include doubling

3965-533: The restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g., y = 2 x + 1 (a line), or x + y = 7 (a circle). Also in the 17th century, Girard Desargues , motivated by the theory of perspective , introduced the concept of idealized points, lines, and planes at infinity. The result can be considered as a type of generalized geometry, projective geometry , but it can also be used to produce proofs in ordinary Euclidean geometry in which

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4030-413: The right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). Thales' theorem , named after Thales of Miletus states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle. Cantor supposed that Thales proved his theorem by means of Euclid Book I, Prop. 32 after

4095-428: The right-angle property of the 3-4-5 triangle) were used long before they were proved formally. The fundamental types of measurements in Euclidean geometry are distances and angles, both of which can be measured directly by a surveyor. Historically, distances were often measured by chains, such as Gunter's chain , and angles using graduated circles and, later, the theodolite . An application of Euclidean solid geometry

4160-420: The shape of a parallelogram with 45º and 135º angles, divided by creases into two diagonal tabs at the ends and two corresponding pockets within the inscribed center square. The system can build a wide range of three-dimensional geometric forms by docking these tabs into the pockets of adjacent units. Three interconnected Sonobe units will form an open-bottomed triangular pyramid with an equilateral triangle for

4225-416: The three medians of ABC intersect), and AL is one of the extended medians of ABC with L lying on the circumcircle of ABC , then BGCL is a parallelogram. Varignon's theorem holds that the midpoints of the sides of an arbitrary quadrilateral are the vertices of a parallelogram, called its Varignon parallelogram . If the quadrilateral is convex or concave (that is, not self-intersecting), then

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