City Of Geometry
( GEOMETROCITY )
Geometry is a branch of mathematics that studies the sizes, shapes,
positions angles, and dimensions of things. Flat shapes like squares,
circles, and triangles are a part of flat geometry and are called 2D
shapes. These shapes have only 2 dimensions, the length and the width .
Solid objects are also known as 3D objects having the third dimension
of height or depth. The earliest recorded beginnings of geometry can be
traced to ancient Mesopotamia and Egypt in the 2nd millennium BC.
A city made of math is called city of geometry or
GEOMETROCITY .The objective of this project is to create a city by
using learned geometry skills and concepts such as: angles, coordinates,
2D shapes, 3D shapes, area, perimeter, volume, transformation, and
more.
Topics Can be :
The arbelos ( The region bounded by three suitably placed
semicircles — studied by Archimedes)
Collinearity of points (Simson’s Theorem and others)*
Concurrrency of lines (Ceva’s Theorem and others)*
Conic Sections (They have many interesting properties and
applications.)*
Constructing geometric figures from string (Many curved solids
actually have lots of straight lines on them, and so models of them
can be constructed of string or wire.)*
Constructing regular polygons (Lots of history, and connections
with number theory.)
Constructing triangles (Various combinations of data, for example
the lengths of two sides and the altitude to one of these sides,
determine a triangle. But how do you construct the triangle given
these data?)*
Constructions with compass only
Curves of constant width
Cycloids and related curves
Dissecting geometric figures (Figures can sometimes be dissected
and rearranged to form other figures. Also, some figures can be
dissected into congruent parts — sometimes even congruent to
the original figure.)
Duality theorems (Sometimes theorems about lines and points are
true if the roles of the lines and points are interchanged.)
Duplicating the cube (It is not possible to construct, with just ruler
and compass, a cube having twice the volume of a given cube.
Why not? Can this be done by other means?)
Euclid’s Elements
Euler’s formula (It relates the number of vertices, edges and faces
of a polyhedron.)
Fivefold symmetry (It occurs frequently in flowers, fruits, and
molecules, but not in crystals. However, it recently was discovered
in "quasicrystals".)
Fractal geometry*
Geodesic structures
Geometric constructions by paper folding (Many ruler-and-
compass type constructions can be done this way, as well as
constructing curves by folding lots of tangents to them.)
Geometry in African art
Geometry in ancient China
Geometry in ancient India
Geometry in ancient Japan
Geometry in archeology
Geometry in architecture
Geometry in astronomy (Shapes of orbits of heavenly bodies;
astronomical calculations)
Geometry in higher dimensions (Four --- and beyond?)
Geometry in machinery (Related to linkages.)
Geometry of animal form and function.
Geometry of billiards*
Geometry of crystals
Geometry of Escher prints (Overlaps with tilings, wallpaper
patterns)
Geometry of Galileo
Geometry of robotics
The geometry of solar energy devices. (Overlaps with geometry of
optics, conic sections)
Geometry of textiles (Geometry occurs both in printed or woven
patterns and in the knitting process. Topic overlaps with knots,
patterns.)
Geometry of the kaleidoscope (Related to tilings and optics.) *
The Golden Ratio in geometry
Inversion (A way of transforming the plane so circles and lines go
to circles and lines, but not necessarily in that order. Related to
complex numbers and linkages.)*
Linkages and geometry (Linkages are mechanical devices that do
things like convert straight line motion to circular motion. Related
to geometry in machinery and to inversion.)
Spherical geometry
Spirals (Overlaps with Golden Section, Fractals)*
Squaring the circle (Can you construct a square whose area is the
same as a given circle? Not by ruler and compass. Are there other
means?)
Topological surfaces
Trisecting the angle. (This is not possible by straightedge and
compass, but is -- to some extent — by other means.)
Wallpaper patterns (Patterns that repeat in two directions)
Vocabulary used :
Parallel lines
Perpendicular Lines
Intersecting Lines
Transversal Lines
Obtuse Angle
Acute Angle
Right Angle
Supplementary Angle
Vertical Angle
Alternate interior angle
Alternate exterior angle
Consecutive interior angle
