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SET C

Greek Geometry

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Euclid in Alexandria

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Video of Euclid in Alexandria

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Deductive method1. The ancient Greeks considered geometry as just one of

several sciences, and held the theorems of geometry on par withscientific facts. As such, they developed and used the logical-

deductive method as a means of avoiding error, and for structuringand communicating knowledge.

2. An "axiom", in classical terminology, referred to a self-evidentassumption common to many branches of science. A good examplewould be the assertion that When an equal amount is taken fromequals, an equal amount results.

3. At the foundation of the various sciences lay certain additionalhypotheses which were accepted without proof. Such a hypothesiswas termed apostulate. While the axioms were common to manysciences, the postulates of each particular science were different.Their validity had to be established by means of real-worldexperience. Indeed, Aristotle warns that the content of a sciencecannot be successfully communicated, if the learner is in doubtabout the truth of the postulates.

4. The classical approach is well illustrated by Euclid's Elements, wherea list of postulates is given (common-sensical geometric facts drawnfrom our experience), followed by a list of "common notions" (verybasic, self-evident assertions).

http://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Geometry

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Deductive method1. The deductive method whereby conclusions follow from

premises through the application of sound arguments

(syllogisms, rules of inference), was developed by theancient Greeks, and has become the core principle ofmodern mathematics.

2. Tautologies excluded, nothing can be deduced if nothing isassumed. Axioms and postulates are the basic assumptionsunderlying a given body of deductive knowledge. They areaccepted without demonstration. All other assertions(theorems) must be proven with the aid of these basicassumptions.

3. However, the interpretation of mathematical knowledge haschanged from ancient times to the modern, and

consequently the terms axiom andpostulate hold a slightlydifferent meaning for the present day mathematician, thanthey did for Aristotle and Euclid.

http://en.wikipedia.org/wiki/Syllogismshttp://en.wikipedia.org/wiki/Syllogismshttp://en.wikipedia.org/wiki/Tautology_(logic)http://en.wikipedia.org/wiki/Theoremhttp://en.wikipedia.org/wiki/Theoremhttp://en.wikipedia.org/wiki/Aristotlehttp://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Aristotlehttp://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Euclidhttp://en.wikipedia.org/wiki/Aristotlehttp://en.wikipedia.org/wiki/Theoremhttp://en.wikipedia.org/wiki/Tautology_(logic)http://en.wikipedia.org/wiki/Syllogisms

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Geometric Problems of Antiquity

The Greek problems of antiquity were aset of geometric problems whose

solution was sought using

only compass and straightedge:1. circle squaring.

2. cube duplication.

3. angle trisection.

http://mathworld.wolfram.com/Compass.htmlhttp://mathworld.wolfram.com/Straightedge.htmlhttp://mathworld.wolfram.com/CircleSquaring.htmlhttp://mathworld.wolfram.com/CubeDuplication.htmlhttp://mathworld.wolfram.com/AngleTrisection.htmlhttp://mathworld.wolfram.com/AngleTrisection.htmlhttp://mathworld.wolfram.com/CubeDuplication.htmlhttp://mathworld.wolfram.com/CircleSquaring.htmlhttp://mathworld.wolfram.com/Straightedge.htmlhttp://mathworld.wolfram.com/Compass.html

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Circle Squaring

1. Construct a square equal in area to a circle using

only a straightedge and compass. This was one of

the three geometric problems of antiquity, and

was perhaps first attempted by Anaxagoras. It

was finally proved to be an impossible problemwhen pi was proven to be transcendental by

Lindemann in 1882.

2. While the circle cannot be squared in Euclideanspace, it can in Gauss-Bolyai-Lobachevsky

Space (Gray 1989).

http://mathworld.wolfram.com/Square.htmlhttp://mathworld.wolfram.com/Area.htmlhttp://mathworld.wolfram.com/Circle.htmlhttp://mathworld.wolfram.com/Straightedge.htmlhttp://mathworld.wolfram.com/Compass.htmlhttp://mathworld.wolfram.com/GeometricProblemsofAntiquity.htmlhttp://mathworld.wolfram.com/Pi.htmlhttp://mathworld.wolfram.com/TranscendentalNumber.htmlhttp://mathworld.wolfram.com/EuclideanSpace.htmlhttp://mathworld.wolfram.com/EuclideanSpace.htmlhttp://mathworld.wolfram.com/Gauss-Bolyai-LobachevskySpace.htmlhttp://mathworld.wolfram.com/Gauss-Bolyai-LobachevskySpace.htmlhttp://mathworld.wolfram.com/Gauss-Bolyai-LobachevskySpace.htmlhttp://mathworld.wolfram.com/Gauss-Bolyai-LobachevskySpace.htmlhttp://mathworld.wolfram.com/Gauss-Bolyai-LobachevskySpace.htmlhttp://mathworld.wolfram.com/Gauss-Bolyai-LobachevskySpace.htmlhttp://mathworld.wolfram.com/Gauss-Bolyai-LobachevskySpace.htmlhttp://mathworld.wolfram.com/Gauss-Bolyai-LobachevskySpace.htmlhttp://mathworld.wolfram.com/Gauss-Bolyai-LobachevskySpace.htmlhttp://mathworld.wolfram.com/EuclideanSpace.htmlhttp://mathworld.wolfram.com/EuclideanSpace.htmlhttp://mathworld.wolfram.com/TranscendentalNumber.htmlhttp://mathworld.wolfram.com/Pi.htmlhttp://mathworld.wolfram.com/GeometricProblemsofAntiquity.htmlhttp://mathworld.wolfram.com/Compass.htmlhttp://mathworld.wolfram.com/Straightedge.htmlhttp://mathworld.wolfram.com/Circle.htmlhttp://mathworld.wolfram.com/Area.htmlhttp://mathworld.wolfram.com/Square.html

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Circle Squaring

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Video of Circle Squaring

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Cube DuplicationThere is a simple neusis construction using a marked ruler for a length

which is the cube root of 2 times another length.[7]Mark a ruler with the given length, this will eventually be GH.

1. Construct an equilateral triangle with the given length as side.

2. Extend AB an equal amount again to D.

3. Extend the line BC forming the line CE.

4. Extend the line DC forming the line CF

5. Place the marked ruler so it goes through A and one end G of themarked length falls on CF and the other end of the marked length

falls on ray CE. Thus GH is the given length.

6. The AG is the given length times the cube root of 2.

http://en.wikipedia.org/wiki/Neusis_constructionhttp://en.wikipedia.org/wiki/Doubling_the_cubehttp://en.wikipedia.org/wiki/Doubling_the_cubehttp://en.wikipedia.org/wiki/Neusis_constructionhttp://en.wikipedia.org/wiki/Neusis_constructionhttp://en.wikipedia.org/wiki/Neusis_construction

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Angle Trisection1. Angle trisection is the division of an arbitraryangle into three

equal angles. It was one of the three geometric problems of

antiquity for which solutions usingonly compass and straightedge were sought. The problem wasalgebraically proved impossible by Wantzel (1836).

2. Although trisection is not possible for a general angle using aGreek construction, there are some specific angles, suchas and radians ( and , respectively), which can be trisected.

Furthermore, some angles are geometrically trisectable, butcannot be constructed in the first place, such as (Honsberger1991). In addition, trisection of an arbitrary angle canbeaccomplished using a markedruler (a Neusis construction) asillustrated above (Courant and Robbins 1996).

3. An angle can also be divided into three (or any whole number) ofequal parts using the quadratrix of Hippias or trisectrix.

http://mathworld.wolfram.com/Angle.htmlhttp://mathworld.wolfram.com/GeometricProblemsofAntiquity.htmlhttp://mathworld.wolfram.com/GeometricProblemsofAntiquity.htmlhttp://mathworld.wolfram.com/GeometricProblemsofAntiquity.htmlhttp://mathworld.wolfram.com/GeometricProblemsofAntiquity.htmlhttp://mathworld.wolfram.com/GeometricProblemsofAntiquity.htmlhttp://mathworld.wolfram.com/GeometricProblemsofAntiquity.htmlhttp://mathworld.wolfram.com/Angle.htmlhttp://mathworld.wolfram.com/GeometricProblemsofAntiquity.htmlhttp://mathworld.wolfram.com/GeometricProblemsofAntiquity.htmlhttp://mathworld.wolfram.com/Compass.htmlhttp://mathworld.wolfram.com/Compass.htmlhttp://mathworld.wolfram.com/Straightedge.htmlhttp://mathworld.wolfram.com/Compass.htmlhttp://mathworld.wolfram.com/Straightedge.htmlhttp://mathworld.wolfram.com/Angle.htmlhttp://mathworld.wolfram.com/Angle.htmlhttp://mathworld.wolfram.com/Angle.htmlhttp://mathworld.wolfram.com/Ruler.htmlhttp://mathworld.wolfram.com/NeusisConstruction.htmlhttp://mathworld.wolfram.com/Angle.htmlhttp://mathworld.wolfram.com/QuadratrixofHippias.htmlhttp://mathworld.wolfram.com/Trisectrix.htmlhttp://mathworld.wolfram.com/Trisectrix.htmlhttp://mathworld.wolfram.com/WholeNumber.htmlhttp://mathworld.wolfram.com/QuadratrixofHippias.htmlhttp://mathworld.wolfram.com/Trisectrix.htmlhttp://mathworld.wolfram.com/Trisectrix.htmlhttp://mathworld.wolfram.com/QuadratrixofHippias.htmlhttp://mathworld.wolfram.com/QuadratrixofHippias.htmlhttp://mathworld.wolfram.com/QuadratrixofHippias.htmlhttp://mathworld.wolfram.com/WholeNumber.htmlhttp://mathworld.wolfram.com/Angle.htmlhttp://mathworld.wolfram.com/NeusisConstruction.htmlhttp://mathworld.wolfram.com/NeusisConstruction.htmlhttp://mathworld.wolfram.com/NeusisConstruction.htmlhttp://mathworld.wolfram.com/Ruler.htmlhttp://mathworld.wolfram.com/Angle.htmlhttp://mathworld.wolfram.com/Angle.htmlhttp://mathworld.wolfram.com/Straightedge.htmlhttp://mathworld.wolfram.com/Compass.htmlhttp://mathworld.wolfram.com/GeometricProblemsofAntiquity.htmlhttp://mathworld.wolfram.com/GeometricProblemsofAntiquity.htmlhttp://mathworld.wolfram.com/Angle.htmlhttp://mathworld.wolfram.com/Angle.html

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Angle Trisection

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Video of Angle Trisection

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Regular Polyhedron

1. A polyhedron is said to be regular if its face and vertexfigures are regular (not necessarily convex) polygons(Coxeter 1973, p. 16). Using this definition, there are atotal of nine regular polyhedra,

2. five being the convexPlatonic solids and four beingthe concave (stellated) Kepler-Poinsot solids.

3. However, the term "regular polyhedra" is sometimesused to refer exclusively to the convexPlatonic solids.

4. It can be proven that only nine regular solids (in theCoxeter sense) exist by noting that a possible regularpolyhedron must satisfy

http://mathworld.wolfram.com/Polyhedron.htmlhttp://mathworld.wolfram.com/RegularPolygon.htmlhttp://mathworld.wolfram.com/Convex.htmlhttp://mathworld.wolfram.com/Convex.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/Concave.htmlhttp://mathworld.wolfram.com/Kepler-PoinsotSolid.htmlhttp://mathworld.wolfram.com/Convex.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/Convex.htmlhttp://mathworld.wolfram.com/Kepler-PoinsotSolid.htmlhttp://mathworld.wolfram.com/Kepler-PoinsotSolid.htmlhttp://mathworld.wolfram.com/Kepler-PoinsotSolid.htmlhttp://mathworld.wolfram.com/Kepler-PoinsotSolid.htmlhttp://mathworld.wolfram.com/Kepler-PoinsotSolid.htmlhttp://mathworld.wolfram.com/Concave.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/Convex.htmlhttp://mathworld.wolfram.com/Convex.htmlhttp://mathworld.wolfram.com/RegularPolygon.htmlhttp://mathworld.wolfram.com/Polyhedron.html

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ConvexPlatonic Solids

http://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/Convex.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/Convex.htmlhttp://mathworld.wolfram.com/PlatonicSolid.html

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ConvexPlatonic Solids

http://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/Convex.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/PlatonicSolid.htmlhttp://mathworld.wolfram.com/Convex.htmlhttp://mathworld.wolfram.com/PlatonicSolid.html

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Characteristics

Symmetry

The regular polyhedra are the most symmetrical ofall the polyhedra. They lie in just three symmetry

groups, which are named after them:1. Tetrahedral

2. Octahedral (or cubic)

3. Icosahedral (or dodecahedral)

Any shapes with icosahedral or octahedralsymmetry will also contain tetrahedral symmetry.

http://en.wikipedia.org/wiki/Symmetry_grouphttp://en.wikipedia.org/wiki/Symmetry_grouphttp://en.wikipedia.org/wiki/Symmetry_grouphttp://en.wikipedia.org/wiki/Symmetry_group

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Characteristics

Euler characteristic

The five Platonic solids have an Euler

characteristic of 2. Some of the regular stars

have a different value.

http://en.wikipedia.org/wiki/Euler_characteristichttp://en.wikipedia.org/wiki/Euler_characteristichttp://en.wikipedia.org/wiki/Kepler-Poinsot_polyhedrahttp://en.wikipedia.org/wiki/Kepler-Poinsot_polyhedrahttp://en.wikipedia.org/wiki/Euler_characteristichttp://en.wikipedia.org/wiki/Euler_characteristic

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Thank you

The End