Electron-pairs repel each other and try to stay as far apart as possible. The valence shell electron pair repulsion ( VSEPR ) model that we describe here focuses on the bonding and nonbonding electron pairs present in the outermost (valence) shell of an atom that connects with two or more other atoms. Like all electrons, these occupy regions of space which we can visualize as electron clouds — regions of negative electric charge, also known as orbitals — whose precise character can be left to more detailed theories. The covalent model of chemical bonding assumes that the electron pairs responsible for bonding are concentrated into the region of apace between the bonded atoms. The fundamental idea of VSEPR thoery is that these regions of negative electric charge will repel each other, causing them (and thus the chemical bonds that they form) to stay as far apart as possible. Thus the two electron clouds contained in a simple triatomic molecule AX2 will extend out in opposite directions; an angular separation of 180° places the two bonding orbitals as far away from each other they can get. We therefore expect the two chemical bonds to extend in opposite directions, producing a linear molecule.

If the central atom also contains one or more pairs of nonbonding electrons, these additional regions of negative charge will behave very much like those associated with the bonded atoms. The orbitals containing the various bonding and nonbonding pairs in the valence shell will extend out from the central atom in directions that minimize their mutual repulsions. If the central atom possesses partially occupied d-orbitals, it may be able to accommodate five or six electron pairs, forming what is sometimes called an expanded octet. 2 Coordination with two or three electron pairs Linear molecules As we stated above, a simple triatomic molecule of the type AX2 has its two bonding orbitals 180° apart, producing a molecule that we describe as having linear geometry. Examples of triatomic molecules for which VSEPR theory predicts a linear shape are BeCl2 (which, you will notice, doesn't possess enough electrons to conform to the octet rule) and CO2. If you write out the electron dot formula for carbon dioxide, you will see that the C-O bonds are double bonds.

This makes no difference to VSEPR theory; the central carbon atom is still joined to two other atoms, and the electron clouds that connect the two oxygen atoms are 180° apart. Trigonal molecules In an AX3 molecule such as BF3, there are three regions of electron density extending out from the central atom. The repulsion between these will be at a minimum when the angle between any two is (360° ÷ 3) = 120°. This requires that all four atoms be in the same plane; the resulting shape is called trigonal planar 3 Tetrahedral coordination Methane, CH4, contains a carbon atom bonded to four hydrogens. What bond angle would lead to the greatest possible separation between the electron clouds associated with these bonds? In analogy with the preceding two cases, where the bond angles were 360°/2=180° and 360°/3=120°, you might guess 360°/4=90°; if so, you would be wrong.

The latter calculation would be correct if all the atoms were constrained to be in the same plane (we will see cases where this happens later), but here there is no such restriction. Consequently, the four equivalent bonds will point in four geometrically equivalent directions in three dimensions corresponding to the four corners of a tetrahedron centered on the carbon atom. The angle between any two bonds will be 109.5°. This is called tetrahedral coordination . This is the most important coordination geometry in Chemistry: it is imperative that you be able to sketch at least a crude perspective view of a tetrahedral molecule. It is interesting to note that the tetrahedral coordination of carbon in most of its organic compounds was worked out in the nineteenth century on purely geometrical grounds and chemical evidence, long before direct methods of determining molecular shapes were developed. For example, it was noted that there is only one dichloromethane, CH2Cl2. If the coordination around the carbon were square, then there would have to be two isomers of CH2Cl2, as shown in the pair of structures here.

The distances between the two chlorine atoms would be different, giving rise to differences in physical properties would allow the two isomers to be distinguished and separated. The existence of only one kind of CH2Cl2 molecule means that all four positions surrounding the carbon atom are geometrically equivalent, which requires a tetrahedral coordination geometry. If you study the tetrahedral figure closely, you may be able to convince yourself that it represents the connectivity shown on both of the "square" structures at the top. A three-dimensional ball-and-stick mechanical model would illustrate this very clearly. Tetrahedrally-coordinated carbon chains Carbon atoms are well known for their tendency to link together to form the millions of organic molecules that are known. We can work out the simpler hydrocarbon chains by looking at each central atom separately. Thus the hydrocarbon ethane is essentially two CH3 tetrahedra joined end-to-end. Similar alkane chains having the general formula H3C–(CH2)n–CH3 (or CnH2n+2) can be built up; a view of pentane, C5H12, is shown below.

Notice that these "straight chain hydrocarbons" (as they are often known) have a carbon "backbone" structure that is not really straight, as is illustrated by the zig-zag figure that is frequently used to denote hydrocarbon structures. is the same as the molecular geometry . The application of VSEPR theory then reduces to the simple problem of naming (and visualizing) the geometric shapes associated with various numbers of points surrounding a central point (the central atom) at the greatest possible angles. Both classes of geometry are named after the shapes of the imaginary geometric figures (mostly regular solid polygons) that would be centered on the central atom and would have an electron pair at each vertex. If one or more of the electron pairs surrounding the central atom is not shared with a neighboring atom (that is, if it is a lone pair), then the molecular geometry is simpler than the coordination geometry, and it can be worked out by inspecting a sketch of the coordination geometry figure.

Tetrahedral coordination with lone pairs In the examples we have discussed so far, the shape of the molecule is defined by the coordination geometry; thus the carbon in methane is tetrahedrally coordinated, and there is a hydrogen at each corner of the tetrahedron, so the molecular shape is also tetrahedral. It is common practice to represent bonding patterns by "generic" formulas such as AX4, AX2E2, etc., in which "X" stands for bonding pairs and "E" denotes lone pairs. (This convention is known as the " AXE method ") The bonding geometry will not be tetrahedral when the valence shell of the central atom contains nonbonding electrons, however. The reason is that the nonbonding electrons are also in orbitals that occupy space and repel the other orbitals. This means that in figuring the coordination number around the central atom, we must count both the bonded atoms and the nonbonding pairs. The water molecule: AX2E2 In the water molecule, the central atom is O, and the Lewis electron dot formula predicts that there will be two pairs of nonbonding electrons.

The oxygen atom will therefore be tetrahedrally coordinated, meaning that it sits at the center of the tetrahedron as shown below. Two of the coordination positions are occupied by the shared electron-pairs that constitute the O–H bonds, and the other two by the non-bonding pairs. Thus although the oxygen atom is tetrahedrally coordinated, the bonding geometry (shape) of the H2O molecule is described as bent. There is an important difference between bonding and non-bonding electron orbitals. Because a nonbonding orbital has no atomic nucleus at its far end to draw the electron cloud toward it, the charge in such an orbital will be concentrated closer to the central atom. As a consequence, nonbonding orbitals exert more repulsion on other orbitals than do bonding orbitals. Thus in H2O, the two nonbonding orbitals push the bonding orbitals closer together, making the H–O–H angle 104.5° instead of the tetrahedral angle of 109.5°.

Although the water molecule is electrically neutral, it is not electrically uniform; the non-bonding electrons create a higher concentration of negative charge (blue color) at the oxygen end, making the hydrogen side relatively positive (red). This image was produced by a computer simulation based on the more complete molecular orbital model that we describe in the next lesson. This charge unbalance gives rise to many of the so-called anomalous properties of water . Ammonia: AX3E Computer-generated image of NH3 molecule showing electrostatic potential (red=+, blue=–.) [ source ] The electron-dot structure of NH3 places one pair of nonbonding electrons in the valence shell of the nitrogen atom. This means that there are three bonded atoms and one lone pair, for a coordination number of four around the nitrogen, the same as occurs in H2O. We can therefore predict that the three hydrogen atom will lie at the corners of a tetrahedron centered on the nitrogen atom. The lone pair orbital will point toward the fourth corner of the tetrahedron, but since that position will be vacant, the NH3 molecule itself cannot be tetrahedral. Instead, it assumes a pyramidal shape.

More precisely, the shape is that of a trigonal pyramid (i.e., a pyramid having a triangular base). The hydrogen atoms are all in the same plane, with the nitrogen above (or below, or to the side; molecules of course dont know anything about above or below!) The fatter orbital containing the non-bonding electrons pushes the bonding orbitals together slightly, making the H–N–H bond angles about 107°. 4 Central atoms with five bonds Compounds of the type AX5 are formed by some of the elements in Group 15 of the periodic table; PCl5 and AsF5 are examples. In what directions can five electron pairs arrange themselves in space so as to minimize their mutual repulsions? In the cases of coordination numbers 2, 3, 4, and 6, we could imagine that the electron pairs distributed themselves as far apart as possible on the surface of a sphere; for the two higher numbers, the resulting shapes correspond to the regular polyhedron having the same number of sides. The problem with coordination number 5 is that there is no such thing as a regular polyhedron with five vertices.

In 1758, the great mathematian Euler proved that there are only five regular convex polyhedra, known as the platonic solids : tetrahedron (4 triangular faces), octahedron (6 triangular faces), icosahedron (20 triangular faces), cube (6 square faces), and dodecahedron (12 pentagonal faces). Chemical examples of all are known; the first icosahedral molecule, LaC60 (in which the La atom has 20 nearest C neighbors) was prepared in 1986. Besides the five regular solids, there can be 15 semi-regular isogonal solids in which the faces have different shapes, but the vertex angles are all the same. These geometrical principles are quite important in modern structural chemistry. The shape of PCl5 and similar molecules is a trigonal bipyramid . This consists simply of two triangular-base pyramids joined base-to-base. Three of the chlorine atoms are in the plane of the central phosphorus atom (equatorial positions), while the other two atoms are above and below this plane (axial positions).

Equatorial and axial atoms have different geometrical relationships to their neighbors, and thus differ slightly in their chemical behavior. In 5-coordinated molecules containing lone pairs, these non-bonding orbitals (which you will recall are closer to the central atom and thus more likely to be repelled by other orbitals) will preferentially reside in the equatorial plane. This will place them at 90° angles with respect to no more than two axially-oriented bonding orbitals. Using this reasoning, we can predict that an AX4E molecule (that is, a molecule in which the central atom A is coordinated to four other atoms X and to one nonbonding electron pair) such as SF4 will have a see-saw shape; substitution of more nonbonding pairs for bonded atoms reduces the triangular bipyramid coordination to even simpler molecular shapes, as shown below. 5 Octahedral coordination Just as four electron pairs experience the minimum repulsion when they are directed toward the corners of a tetrahedron, six electron pairs will try to point toward the corners of an octahedron .

An octahedron is not as complex a shape as its name might imply; it is simply two square-based pyramids joined base to base. You should be able to sketch this shape as well as that of the tetrahedron. The shaded plane shown in this octahedrally-coordinated molecule is only one of three equivalent planes defined by a four-fold symmetry axis. All the ligands are geometrically equivalent; there are no separate axial and equatorial positions in an AX6 molecule. Click here to see an image that shows the symmetry of the octahedron in more detail. At first, you might think that a coordination number of six is highly unusual; it certainly violates the octet rule, and there are only a few molecules (SF6 is one) where the central atom is hexavalent. It turns out, however, that this is one of the most commonly encountered coordination numbers in inorganic chemistry. There are two main reasons for this: Many transition metal ions form coordinate covalent bonds with lone-pair electron donor atoms such as N (in NH3) and O (in H2O).

Since transition elements can have an outer configuration of d10s2, up to six electron pairs can be accommodated around the central atom. A coordination number of 6 is therefore quite common in transition metal hydrates, such as Fe(H2O)63+. Although the central atom of most molecules is bonded to fewer than six other atoms, there is often a sufficient number of lone pair electrons to bring the total number of electron pairs to six. Octahedral coordination with lone pairs There are well known examples of 6-coordinate central atoms with 1, 2, and 3 lone pairs. Thus all three of the molecules whose shapes are depicted below possess octahedral coordination around the central atom. Note also that the orientation of the shaded planes shown in the two rightmost images are arbitrary; since all six vertices of an octahedron are identical, the planes could just as well be drawn in any of the three possible vertical orientations. 6 Summary of VSEPR theory The VSEPR model is an extraordinarily powerful one, considering its great simplicity. Its application to predicting molecular structures can be summarized as follows: 1.

Electron pairs surrounding a central atom repel each other; this repulsion will be minimized if the orbitals containing these electron pairs point as far away from each other as possible. 2. The coordination geometry around the central atom corresponds to the polyhedron whose number of vertices is equal to the number of surrounding electron pairs (coordination number). Except for the special case of 5, and the trivial cases of 2 and 3, the shape will be one of the regular polyhedra. 3. If some of the electron pairs are nonbonding, the shape of the molecule will be simpler than that of the coordination polyhedron. 4. Orbitals that contain nonbonding electrons are more concentrated near the central atom, and therefore offer more repulsion than bonding pairs to other orbitals. While VSEPR theory is quite good at predicting the general shapes of most molecules, it cannot yield exact details. For example, it does not explain why the bond angle in H2O is 104.5°, but that in H2S is about 90°.

This is not surprising, considering that the emphasis is on electronic repulsions, without regard to the detailed nature of the orbitals containing the electrons, and thus of the bonds themselves. The Valence Shell Electron Repulsion theory was developed in the 1960s by Ronald Gillespie of McMaster University (Hamilton, Ontario, Canada) and Ronald Nyholm (University College, London). It is remarkable that what seems to be a logical extension of the 1916 Lewis shared-electron pair model of bonding took so long to be formulated; it was first presented in the authors' classic article Inorganic Stereochemistry published in the 1957 Chemical Society of London Quarterly Reviews (Vol 11, pg 339). Although it post-dates the more complete quantum mechanical models, it is easy to grasp and within a decade had become a staple of every first-year college chemistry course. What you should be able to do Make sure you thoroughly understand the following essential ideas which have been presented above. Describe the manner in which repulsion beween electron-pairs affects the orientation of the regions that contain them.

Define coordination geometry, and describe the particular geometry associated with electron-pair repulsion between two, three, four, five, or six identical bonding regions. Explain the distinction between coordination geometry and molecular geometry, and provide an illustration based on the structure of water or ammonia. Draw a diagram of a tetrahedral or octahedral molecule. Concept Map Geometry History Home Welcome to the new improved GeomHistory.com web site. This site presents a brief outline of the history of geometry (Ancient, Greek, Medieval, and Modern), with timelines of the major geometers responsible for it's evolution. Click on any person's name or picture for an expanded biography at the History of Mathematics Archive (Univ of St Andrews, Scotland). For the history of geometry, going through our site facilitates accessing the information in the St Andrews Archive. Ancient Geometry (3000 BC - 500 BC) Babylon (2000 BC - 500 BC) The Babylonians replaced the older (4000 BC - 2000 BC) Sumerian civilization around 2000 BC.

The Sumerians had already developed writing (cuniform on clay tablets) and arithmetic (using a base 60 number system). The Babylonians adopted both of these. But, Babylonian math went beyond arithmetic, and devloped basic ideas in number theory, algebra, and geometry. The problems they wanted to solve usually involved construction and land estimation, such as areas and volumes of rectangular objects. Some of their methods were rules that solved specialized quadratic, and even some cubic, equations. But, they didn’t have algebraic notation, and there is no indication that they had logical proofs for the correctness of their rule-based methods. Nevertheless, they knew some special cases of the "Pythagorean Theorem" more than 1000 years before the Greeks (see: Pythagorean Knowledge In Ancient Babylonia and Pythagorus’ theorem in Babylonian mathematics ). Their durable clay tablets have preserved some of their knowledge (better than the fragile Eygptian papyri). Four specific tablets (all from the period 1900 BC - 1600 BC) give a good indication of Babylonian mathematical knowledge: Yale tablet YBC 7289 - shows how to compute the diagonal of a square.

Plimpton 322 - has a table with a list of Pythagorean integer triples. Susa tablet - shows how to find the radius of the circle through the three vertices of an isoceles triangle. Tell Dhibayi tablet - shows how to find the sides of a rectangle with a given area and diagonal. There is no direct evidence that the Greeks had access to this knowledge. But, some Babylonian mathematics was known to the Eygptians; and probably through them, passed on to the Greeks (Thales and Pythagorus were known to have traveled to Egypt). Egypt (3000 BC - 500 BC) The geometry of Egypt was mostly experimentally derived rules used by the engineers of those civilizations. They developed these rules to estimate and divide land areas, and estimate volumes of objects. Some of this was to estimate taxes for landowners. They also used these rules for construction of buildings, most notably the pyramids. They had methods (using ropes to measure lengths) to compute areas and volumes for various types of objects, various triangles, quadrilaterals, circles, and truncated pyramids. Some of their rule-based methods were correct, but others gave approximations.

However, there is no evidence that the Egyptians logically deduced geometric facts and methods from basic principles. And there is no evidence that they knew a form of the "Pythagorean Theorem", though it is likely that they had some methods for constructing right angles. Nevertheless, they inspired early Greek geometers like Thales and Pythagorus. Perhaps they knew more than has been recorded, since most ancient Eygptian knowledge and documents have been lost. The only surviving documents are the Rhind and Moscow papyri . The book Mathematics in the Time of the Pharaohs gives a more detailed analysis of Egyptian mathematics. India (1500 BC - 200 BC) Everything that we know about ancient Indian (Vedic) mathematics is contained in: The Sulbasutras These are appendices to the Vedas, and give rules for constructing sacrificial altars. To please the gods, an altar's measurements had to conform to very precise formula, and mathematical accuracy was very important. It is not historically clear whether this mathematics was developed by the Indian Vedic culture, or whether it was borrowed from the Babylonians.

Like the Babylonians, results in the Sulbasutras are stated in terms of ropes; and "sutra" eventually came to mean a rope for measuring an altar. Ultimately, the Sulbasutras are simply construction manuals for some basic geometric shapes. It is noteworthy, though, that all the Sulbasutras contain a method to square the circle (one of the infamous Greek problems) as well as the converse problem of finding a circle equal in area to a given square. The main Sulbasutras, named after their authors, are: Baudhayana (800 BC) Baudhayana was the author of the earliest known Sulbasutra. Although he was a priest interested in constructing altars, and not a mathematician, his Sulbasutra contains geometric constructions for solving linear and quadratic equations, plus approximations of (to construct circles) and . It also gives, often approximate, geometric area-preserving transformations from one geometric shape to another. These include transforming a square into a rectangle, an isosceles trapezium, an isosceles triangle, a rhombus, and a circle, and finally transforming a circle into a square.

Further, he gives the special case of the “Pythagorean theorem” for the diagonal of a square, and also a method to derive “Pythagorian triples”. But he also has a construction (for a square with the same area as a rectangle) that implies knowing the more general “Pythagorian theorem”. Some historians consider the Baudhayana as the discovery of the “Pythagorian theorem” . However, the Baudhayana descriptions are all empirical methods, with no proofs, and were likely predated by the Babylonians. contains approximate constructions of circles from rectangles, and squares from circles, which give an approximation of = 25/8 = 3.125. Apastamba (600-540 BC) considers the problems of squaring the circle, and of dividing a segment into 7 equal parts. It also gives an accurate approximation of = 577 / 408 = 1.414215686, correct to 5 decimal places. states the general case of the Pythagorean theorem for the diagonal of any rectangle. © Copyright 2012 Dan Sunday, 2001 softSurfer. All rights reserved.

greek geometry A Brief History of Greek Geometry A Little Background: The word geometry has its roots in the Greek work geometrein, which means “earth measuring”. Before the time of recorded history, geometry originated out of practical necessity; it was the science of measuring land. Many ancient civilizations (Babylonian, Hindu, Chinese, and Egyptian) possessed geometric information. The first geometrical considerations “had their origin in simple observations stemming from human ability to recognize physical form and to compare shapes and sizes” (Historical Topics, 165). There were many circumstances in which primitive people were forced to take on geometric topics, although it may not have been recognized as such. For instance, man had to learn with situations involving distance, bounding their land, and constructing walls and homes. These types of situations were directly related to the geometric concepts of vertical, parallel, and perpendicular. The geometry of the ancient days was actually just a collection of rule-of-thumb procedures, which were found through experimentation, observation of analogies, guessing, and sometimes even intuition. Basically, geometry in the ancient days allowed for approximate answers, which were usually sufficient for practical purposes. For example, the Babylonians took p to be equal to 3.

It is said that the Babylonians were more advanced than the Egyptians in arithmetic and algebra. They even knew the Pythagorean theorem long before Pythagoras was even born. The Babylonians had an algebraic influence on Greek mathematics. Egyptian geometry was not a science in the way the Greeks viewed geometry. It was more of a grab bag for rules for calculation without any motivation or justification. Sometimes they guessed correctly, but other times they did not. One of their greatest accomplishments was finding the correct formula for the volume of a frustum of a square pyramid. However, they thought that the formula that they had for the area of a rectangle could be applied to any quadrilateral. Primitive people could not escape geometry in the same way that we cannot escape it today. The concept of the curve was found in flowers and the sun, a parabola was represented by tossing an object, and spider webs posed an excellent example of regular polygons. Symmetry could be seen in many living objects, including man, and the idea of volume had to be addressed when constructing a device to hold water. Historical Topics for the Mathematics Classroom calls this type of geometry “subconscious geometry”. This is the type of geometry that very young children experience as they begin to play with objects.

This type of geometry involves concrete objects. Still before the time of recorded history, man began to consider situations that were more hypothetical. They were able to take the knowledge they had learned from observation of concrete objects and come up with general algorithms and procedures to be used in particular cases. This is what Historical Topics for the Mathematics Classroom refers to as “scientific geometry” (166-7). Procedures such as trial and error, induction, and rule-of thumb were being used to discover. This was mainly the geometry of the Babylonians and the Egyptians. Although there is no evidence that they were able to deductively reason geometric facts from basic principles, it is thought that they paved the way for Greek geometry. Geometry remained this way (“scientific”) until the Greek period. A Bit of Greek Geometry (600 BC – 400 AD): To see a chronological outline of the work of Greek geometers, click here. (An entire mathematical chronology can be found by visiting: www-groups.dcs.st-and.ac.uk/~history/Chronology/full.html.) The Greeks worked to transform geometry into something much different than the “scientific geometry” of the people that worked before them.

“The Greeks insisted that geometric fact mush be established, … , by deductive reasoning; …” (Historical Topics, 171). They believed that geometrical truth would be found by studying rather than experimenting. They transformed the former “scientific geometry” into a more “systematic geometry”. Keep in mind that there exist virtually no first-hand sources of early Greek geometry. Hence, the following is based on manuscripts written hundreds of years after this early Greek geometry had been developed. According to these manuscripts, Thales of Miletus was the one who began early Greek geometry in the sixth century B.C. He is noted as one of the first known to indulge himself in deductive methods in geometry. His credited elementary geometrical findings resulted from logical reasoning rather than intuition and experiment. He insisted that geometric statements be established by deductive reasoning rather than by trial and error. He was familiar with the computations recorded from Egyptian and Babylonian mathematics, and he developed his logical geometry by determining which results were correct. The next mentioned great Greek geometer is one who quite possibly studied under Thales of Miletus.

This geometer is Pythagoras , who founded the Pythagorean school, which was “committed to the study of philosophy, mathematics, and natural science” (Historical Topics, 172). In the area of geometry, the members of this school developed the properties of parallel to prove that the sum of any angles of a triangle is equal to two right angles. They also worked with proportion to study similar figures. The deductive side of geometry was further developed during this time. We all think of the Pythagorean Theorem when we think of Pythagoras, however it is important to note that this theorem was used (although it may not have been proved) before his time. As an interesting side note, Pythagoras was regarded as a religious prophet by his contemporaries. He preached the immortality of the soul and reincarnation, and he even organized a brotherhood of believers. This brotherhood had initiation rites, they were vegetarian, and they shared all property. They did, however, differ from other religious groups in one major way. They believed that elevation of the soul and union with God was achieved through the study of music and mathematics. Hippocrates of Chios was one of these students at the Pythagorean school.

It is suggested that he was the first to attempt “a logical presentation of geometry in the form of a single chain of propositions …” (Historical Topics, 172). He is credited to writing the first “Elements of Geometry” where he included geometric solutions to quadratic equations and some of the first methods of integration. He studied the problem of squaring a circle and squaring a lune . He also was the first to show that the ratio of the areas of two circles equals the ratio of the squares of the circles’ radii ( Although Plato did not make any major mathematical discoveries himself, he did emphasize the idea of proof. He insisted on accuracy, which helped pave the way for Euclid. It is correct to say that almost every significant geometrical development can be traced back to three outstanding Greek geometers: Euclid, Archimedes, and Apollonis. Euclid collected the theorems of Pythagoras, Hippocrates, and others into a work called “The Elements”. ( Euclid is the most widely read author in the history of mankind. The teaching of geometry has been dominated by Euclid’s approach to the subject. In fact, Euclid’s axiomatic method is the prototype for all “pure mathematics”.

By “pure”, it is meant that all statements can be verified through reasoning of demonstrations; no physical experiments are necessary. Typically, the next mentioned Greek mathematician is regarded as the greatest Greek mathematician by geometryalgorithms.com. His name was Archimedes of Syracuse . He had many mathematical accomplishments as well as being the inventor of the screw, the pulley, the lever, and other mechanical devices. He perfected integration using the method of exhaustion discovered my Eudoxus, and he was able to find the areas and volumes of many objects. Inscribed on his tomb was the result he found that the volume of a sphere is two-thirds the volume of its circumscribed cylinder. ( Apollonius was an astronomer who had his mathematical bid to fame in his work entitled Conic Sections. It is this great Greek geometer who provided us with the terms “ellipse,” “parabola,” and “hyperbola.” He is also accredited with showing how to construct a circle which is tangent to three objects. His approximation of π was even closer than that of Archimedes. ( Last, but certainly not least, Hypatia of Alexandria was the first woman to substantially contribute to mathematics.

She studied under her father and assisted him in writing a new version of Euclid’s Elements. She also wrote commentaries on other great Greek geometer’s works. She was the “first woman in history recognized as a professional geometer and mathematician” ( Greenberg, Marvin J. Euclidean and Non-Euclidean Geometries: Development and History. 3rd ed. New York: W.H. Freeman and Company, 1993. 6-19. Geometric - definition of geometric by The Free Dictionary Geometric - definition of geometric by The Free Dictionary (jē′ə-mĕt′rĭk) also ge·o·met·ri·cal (-rĭ-kəl) adj. a. Of or relating to geometry and its methods and principles. b. Increasing or decreasing in a geometric progression. 2. Using simple geometric forms such as circles and squares in design and decoration. 3. Of or relating to properties in algebraic geometry involving algebraically closed fields. ge′o·met′ri·cal·ly adv. geometric adj 1. (Mathematics) of, relating to, or following the methods and principles of geometry 2.

(Mathematics) consisting of, formed by, or characterized by points, lines, curves, or surfaces: a geometric figure. 3. (Art Terms) (of design or ornamentation) composed predominantly of simple geometric forms, such as circles, rectangles, triangles, etc ˌgeoˈmetrically adv adj. Also, ge`o•met′ri•cal. 1. of or pertaining to geometry or to its principles. 2. resembling, employing, or characterized by the simple rectilinear or curvilinear lines or figures used in geometry. n. 3. Often, geometrics. a geometric pattern, design, etc. [1620–30; < Latin < Greek] ge`o•met′ri•cal•ly, adv. geometric 1. (') A unit of measure for plane angles. 1' = 1⁄60°. 2. (") A measure of plane angle equal to 1⁄360°. ThesaurusAntonymsRelated WordsSynonymsLegend: Adj. 1.

geometric - characterized by simple geometric forms in design and decoration; "a buffalo hide painted with red and black geometric designs" geometrical beaux arts , fine arts - the study and creation of visual works of art nonrepresentational - of or relating to a style of art in which objects do not resemble those known in physical nature 2. geometry (dʒiˈomətri) noun a branch of mathematics dealing with the study of lines, angles etc. He is studying geometry.

geometrie هندسَه геометрия geometria geometrie die Geometrie geometri γεωμετρία geometría geomeetria هندسه geometria géométrie גיאומטרייה रेखा गणित geometrija mértan geometri rúmfræði geometria 幾可学 기하학 geometrija ģeometrija geometri meetkunde geometri geometria جيومټرى، هندسه، دهندسې علم geometria geo­metrie геометрия geometria geometrija geometrija geometri เรขาคณิต geometri 幾何學 геометрія علم ہندسہ hình học 几何学 geometric(al) (dʒiəˈmetrik(əl)) adjective made up of lines, circles etc and with a regular shape. a geometrical design on wallpaper.

geometriese هَنْدَسي геометричен geométrico geometrický geometrisch geometrisk γεωμετρικός geométrico geomeetriline هندسی geometrinen géométrique גיאומטרי ज्यामितीय, गुणोत्तर geometrijski geometriai, mértani geometris rúmfræðilegur geometrico 幾可学の 기하학(상)의 geometrinis ģeometrisks geometri geometrisch geometrisk geometryczny هندسې geométrico geo­me­tric геометрический geometrický geometrijski geometrijski geometrisk ทางเรขาคณิต geometrik 幾何(學)的 геометричний لکیروں ، اشکال وغیرہ کی مدد سے بنایا گیا thuộc hình học 几何（学）的 ˌgeoˈmetrically adverb geometriese هَنْدَسِيّا геомет

рично geometricamente geometricky geometrisch geometriskt γεωμετρικά geométricamente geomeetriliselt بطور هندسی geometrisesti géométriquement מבחינת גיאומטרית ज्यामीत‍ि से geometrijski mértanilag secara geometris rúmfræðilega geometricamente 幾可学的に 기하학적으로 geometriškai ģeometriski secara geometrinya geometrisch geometrisk geometrycznie د هندسی په دوله geometricamente din punct de vedere geometric геометрически geometricky geometrijsko geometrijski geometriskt ทางเรขาคณิต geometrik olarak 幾何(學)上 за законами геометрії شکل و بناوٹ کے لہاظ سے về phương diện hình học 几何（学）上 27 glorious geometric patterns in design | Creative Bloq 27 glorious geometric patterns in design 27 glorious geometric patterns in design An inspiring collection of geometric patterns featuring illustration, branding and more.

Shares Lately, we've noticed a lot of designers using geometric patterns, shapes and styles in their logo designs , vector art and more. Using these shapes, the designs become a simple yet wholly striking work of art channelling influences from the design era of art deco. We've rounded up our favourite examples of geometric patterns and designs featuring geometric shapes. See what you think... 01. Eric Broug Eric Broug's art is inspired by Islamic geometric design Dutch artist and designer Eric Broug discovered Islamic geometric art as a student in Amsterdam, and has been pursuing it ever since. Because the use of figurative images is forbidden in Islamic art, it often uses intricate geometric patterns, created by the repetition, overlapping and interlacing of squares and circles, following mathematical rules. Broug taught himself about Islamic geometric art by trying to deconstruct and recreate its patterns with a compass, ruler, pencil and paper, and today he creates his own Islamic-style designs when he's not travelling the world for an academic publisher. 02.

Jeremy Booth Curiosity demonstrates Jeremy Booth's geometric approach to illustration Born and raised in Louisville, Kentucky, Jeremy Booth is a self-taught designer and illustrator whose style has been described as 'vector noir', with an emphasis on strong angular lines with plenty of bold light and shadow. Much of his work features distinctly geometric elements, as can be seen in one of his recent illustrations, Curiosity. Head to his site to see more examples of his eye-catching work. 03. City of Melbourne This cool design is as multi-faceted as the city itself Bringing a city together through branding is no easy task, especially when the city in question is a diverse as Melbourne, Australia. However that's exactly what branding agency Landor had to achieve with their aesthetic for the City of Melbourne Council. Thanks to a clever geometric design, the chunky 'M' logo is flexible enough to reflect the different aspects and personalities of the city. Accompanied by a broader branding campaign that spreads across print and online platforms, Landor have successfully tied together the city through angular imagery. 04.

Olivia King's packaging This dynamic packaging really complements the jewellery Australian designer, printer and podcaster Olivia King created this beautiful collection of concept packaging, which is suitably called Trigonometry. With a focus on angles and a bright visual identity, this design links up physical good with a digital app. Thanks to its uniform look and feel, this gorgeous design really elevates the value of the products and packaging. Now we want to see it hit the shelves for real! 05. Nancy Mitford covers This series of 8 novels each have a distinctive geometric cover Commissioned by Penguin Random House's in-house design team, New York-based artist Lourdes Sanchez created these soothing geometric covers for new editions of Nancy Mitford books. To take the edge off what could be visually overwhelming covers, these designs were created with watered-down paints that make for a pleasant finish. Each of the eight novels have a different design, ranging from a mosaic effect, diagonal stripes and teardrops. 06. Infinity Koseoglu's work focuses on naturally occuring geometric patterns Based in London, Umit Koseoglu is a digital artist, fine art, and portrait photographer.

His work focuses around the beauty of the natural world, with this geometric offering honing in on Iceland's environments. "Infinity is a body of work created by isolating naturally occuring patterns found within the Icelandic landscape to form tessellating abstract images, which can be infinitely repeated," he explains. You can take a look at more of the images on his website – you can also purchase prints of your favourites! 07. Dogs Ty Dale uses geometric shapes in most of his work Freelancer Ty Dale focuses much of his work using geometric shapes. From hot dogs to actual dog illustrations, his bold use of colours and playful approach means we've fallen head-over-heels for his geometric style. As well as editorial illustrations, he's also produced geometric prints that you'll definitely want to get your hands on. 08. Feedback Cam "I've been working on a webcam-feedback generator, which makes geometric patterns," explains interactive developer Lawrie Cape. "Basically, it's a website, where you point your webcam back at the screen." Producing breathtaking geometric patterns, it's a unique approach to geometric design. Have a go yourself on the website . 09.

Vector animals This geometric vector series is a marvel of inspiration This project from designer Hope Little , which began back in 2012, is a marvel of geometric design. These vector animals have proven so popular that Little has even started taking requests for portraits. "I wanted to steer away from my usual melty, disproportional illustrations and try for something clean and balanced," explains Little. "I started experimenting with shapes, settling on a triangle to keep things clean and simple. I wanted the illustrations to be bright and colourful, so I searched for animals; do to the fact the fur offered a wide variety of patterns and colours." 10. Geometric tattoos More geometric animals here, this time on your skin Combining precise shapes and bold, in-your-face colours, these tattoos are a thing of beauty. The geometric aspect makes them triumph over the usual tattoo offerings, with Russian tattoo artist Sasha Unisex 's creations proving a hit across the world. Operating out of her studio in St. Petersburg, it seems Sasha is booked up for months in advance with just about every tattoo lover wanting a piece of her art on their body – who can blame them? 11.

Spray paintings This series of paintings were created by New York artist Adam Daily This series of beautiful geometric paintings was crafted by New York artist Adam Daily , whose work spans a variety of media and techniques, including painting, photography and collage. These paintings were imagined through a combination of both digital and analog tools and were eventually created by hand, using acrylic on PVC panels and applying paint with a spray gun. 12. Geometric clothes Will you really be wearing these geometric patterns in the future? Icelandic designer Sruli Recht has taken geometric design to a new level with these designs for a range of futuristic fashion. Recht explains the concept as: "The simplified disastery of polygonal geometry - breaking the body down into a pixelated memory." The clothes are made from "walnut wood material on a wool base. Once grown, the wood is deconstructed into pieces, and then attached to a textile base, creating a material that is half wood, half textile, and completely fragmentary." 13.

Justin Maller's Facets Number 141 of 365 is called Handsome II - with good reason In 2013, New York-based designer and illustrator Justin Maller accepted a challenge from his girlfriend to create a new piece of personal artwork every day for 12 months. He accepted, got to work and is now the proud owner of 365 beautiful, geometric-style images that make a series of artwork titled Facets . You can find out more about it in our interview with Maller here . 14. Landshape The poster was created for a Tame Impala gig in Rio Liam Brazier is a freelance illustrator and animator based in London. He creates a wide range of geometric inspired designs for a range of clients including Cartoon Network, Dazed & Confused, and Glastonbury Festival. This poster design was created for a Tame Impala gig in Rio and is cased upon the band's album artwork. 15. KYL21 popsicle A tasty science experiment with these vegan popsicles! Developed and designed by The Science Kitchen , founder and food designer David Marx aims to re-invent food in new and innovative ways.

The KYL21 popsicle is a vegan and lactose-free popsicle that comes in an array of precisely sculpted geometric shapes. 16. Bird Mural Tobias Hall works as Zizzi's in-house art director Only graduating in 2010, designer Tobias Hall has been working closely with UK-based Italian restaurant chain, Zizzi , as their in-house illustrator, designer and art director. Creating a number of bespoke wall paintings, we particularly loved this geometric bird design. 17. Portobello Books A geometric inspired book cover design by Melvin Galapon This geometric inspired book cover design for ' Drugs 2.0 - The Web Revolution That's Changing How The World Gets High ' by Mike Power was created by British designer Melvin Galapon . Currently based in London, he works on a prolific mix of portraits, type, installations, textile and video work, this is just one example of his inspiring work. 18. Motif wine These geometric pattern wine bottles will certainly brighten up your kitchen Designed by Motif with geometric pattern lovers in mind, the selection of wines come with its own unique colour that represent each wine's taste and smell – a perfect motif, if you will. "We believe the time is right.

For a wine without many words," Motif explain. 19. Pattern Booth Patternbooth create a number of geometric inspired pattern designs The patternbooth label is a collection of fresh, abstract and geometric pattern designs created by Cara Holland for interiors, fashion and products. She has a number of inspiring examples on her website but we particularly enjoyed the colour scheme of this one. 20. Matt W Moore Matt W Moore has been painting geometric pattern designs for over half his life We love seeing inspirational examples of street art – often brightening up grey and drab architecture with a lick of paint and a load of creativity. Boston based artist Matt W Moore has been painting on walls for over half his life and this geometric pattern design is just some one example of his incredible work. 21. Geometric Daily An inspiring geometric pattern design is posted up every day on this tumblr Every day, a new geometric minimal composition is created and posted on the geometric daily tumblr . Proving to be one of the best tumblr blogs for designers , it packed full of minimalist inspiration and gorgeous colour schemes. Scroll away! 22. Oh My God These well-known Gods were created for an exhibition in Barcelona This brilliant new series of geometric deities comes straight out of Barcelona-based creative agency Hey Studio .

They were created for the aptly titled 'Oh My God' show at the Mitte-Barcelona space. The modern vector illustrations make us look at these well known characters from a new perspective. 23. Little White Lies This geometric inspired cover even came with 3D glasses Little White Lies have produced some of the most inspiring and beautiful magazine covers of the past few years. Here, they designed a special cover for Paul Thomas Anderson's The Master that featured clever use of geometric shapes. The issue even came with 3D glasses! 24. Newso Freelance graffiti artist Newso paints another bespoke set of stairs Freelance graffiti artist Newso was commissioned to paint another stairway after his work in Boxxed . "Painting stairs is unusual, I wouldn’t have thought I'd have done two sets in one summer," he explains. "I have found the angle of stairs work well with my grid so the triangles fit well in the panels." 25. Sakir Gökçebag Geometric food art that looks good enough to eat Who needs Photoshop when you're this handy with a knife?

Turkish artist Sakir Gökçebag has breathed new life into every day produce such as apples and watermelons and turned them into something quite beautiful. The geometric shapes created have not been digitally remastered or altered in any way! 26. Emily Forgot A delightful geometric poster design by Emily Forgot This artwork was created for the Manchester weekender festival 2013 under the art direction of Modern Designers . It was drawn by illustrator Emily Forgot – the appropriately curious moniker of London based Graphic Artist Emily Alston. 27. Mirage animation This vivid new animation 'Mirage' was created by video and sound designer Frederic Kokott . Using abstract and geometric forms, Kokott brings an unnamed city slowly into focus in the four-minute short. Featuring flat colours and simple shapes, the designer used Adobe Illustrator and After Effects to bring all the elements together. ABOUT THE AUTHOR Geometry in Art & Architecture Unit 5 Slide 5-23: Design at Pompeii Calter photo In the last unit, Number Symbolism we saw that in the ancient world certain numbers had symbolic meaning, aside from their ordinary use for counting or calculating. But each number can be associated with a plane figure, or polygon (Three and the Triangle, for example).

In this unit we'll see that each of these polygons also had symbolic meaning and appear in art motifs and architectural details, and some can be classified as sacred geometry. A polygon is a plane figure bounded by straight lines, called the sides of the polygon. From the Greek poly = many and gon = angle The sides intersect at points called the vertices. The angle between two sides is called an interior angle or vertex angle. Regular Polygons A regular polygon is one in which all the sides and interior angles are equal. Polygons vs. Polygrams A polygram can be drawn by connecting the vertices of a polgon. Pentagon & Pentagram, hexagon & hexagram, octagon & octograms Equilateral Triangle Slide 5-2: Tablet in School of Athens, showing Tetractys Bouleau There are, of course, an infinite number of regular polygons, but we'll just discuss those with sides from three to eight. In this unit we'll cover just those with 3, 5, and 6 sides. We'll start with the simplest of all regular polygons, the equilateral triangle. Sacred Tetractys The Pythagoreans were particularly interested in this polygon because each triangular number forms an equilateral triangle.

One special triangular number is the triangular number for what they called the decad, or ten, the sacred tetractys. Ten is important because it is, of course, the number of fingers. The tetractys became a symbol of the Pythagorean brotherhood. We've seen it before in the School of Athens. Trianglular Architectural Features Slide 5-6: Closeup of wheel Calter photo Slide 5-7: Pompeii Tiling with equilateral triangles Calter photo Tilings or tesselations refers to the complete covering of a plane surface by tiles. There are all sorts of tilings, some of which we'll cover later. For now, lets do the simplest kind, called a regular tiling, that is, tiling with regular polygons. This is opposed to semiregular tilings like the Getty pavement shown here. Slide 5-8: Getty Pavement Calter photo The equilateral triangle is one of the three regular polygons that tile a plane. the other two being the square and hexagon. Hexagon & Hexagram Slide 5-20: Seal on Dollar Bill Calter photo The hexagrarn can also be viewed as two overlapping Pythagorean tetractys.

Joseph Campbell writes; In the Great Seal of the U.S. there are two of these interlocking triangles. We have thirteen points, for our original thirteen states, and six apexes: one above, one below, andfour to thefour quarters. The sense of this might be thalftom above or below, orftom any point of the compass, the creative word may be heard, which is the great thesis of democracy. - The Power of Myth. p.27 Hexagonal Designs in Architecture Hexagonal designs are common in ancient architecture, such as this church window in Quebec. Slide 5-22: Church Window in Quebec Calter photo This marvelous design is at Pompeii. It is made up of a central hexagon surrounded by squares, equilateral triangles, and rhombi. Slide: 5-23. Design at Pompeii Calter photo Slide 5-24: Design on Pisa Duomo Calter photo This hexagram is one of countless designs on the Duomo in Pisa.

Pentagon & Pentagram Slide 5-26: Pentagram from grave marker Calter photo The Pentagram was used as used as a sign of salutaton by the Pythagoreans, its construction supposed to have been a jealously guarded secret. Hippocrates of Chios is reported to have been kicked out of the group for having divulged the construction of the pentagram. The pentagram is also called the Pentalpha, for it can be thought of as constructed of five A's. Euclid's Constructions of the Pentagon Euclid gives two constructions in Book IV, as Propositions 11 & 12. According to the translator T.L. Heath, these methods were probably developed by the Pythagoreans. Medieval Method of Construction Supposedly this construction was one of the secrets of Medieval Mason's guilds. It can be found in Bouleau p. 64. Durer's Construction of the Pentagon Another method of construction is given in Duret's "Instruction in the Measurement with the Compass and Ruler of Lines, Surfaces and Solids," 1525.

Its the same construction as given in Geometria Deutsch, a German book of applied geometry for stonemasons and Golden Ratios in the Pentagram and Pentagon The pentagon and pentagram are also interesting because they are loaded with Golden ratios, as shown in Boles p.48. Golden Triangle Slide 5-28: Emmer, plate F3 Emmer, Michele, Ed. The Visual Mind: Art and Mathematics. Cambridge: MIT Press, 1993. The Golden Triangle A golden triangle also called the sublime triangle, is an isoceles triangle whose ratio of leg to base is the golden ratio. It is also an isoceles triangle whose ratio of base to leg is the golden ratio, so there are two types: Type I, acute, and type II, obtuse. A pentagon can be subdivided into two obtuse and one acute golden triangle. Euclid's Construction Euclid shows how to construct a golden triangle. Book IV, Proposition 10 states, "To construct an isoceles triangle having each of the angles at the base the double of the remaining one." Penrose Tilings

Question: What geometrical shape most closely approximates India?
Answer: Triangle