NonEuclid
is a simulation by Joel Castellanos from
Rice University that allows you to draw lines and
circles in the Hyperbolic Plane. The material below
was developed by him with some material added about
parallel lines.
Many of the items in the "Help" menu present a set
of statements that are theorems in Euclidean Geometry.
Your job is to determine which of the statements are
also theorems in Hyperbolic Geometry.
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Table of
Contents:
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Adjacent Angles
The following statement is a theorem in
Euclidean Geometry:
Euclidean Theorem:
The adjacent angles
formed by a pair of intersecting lines are
supplementary and together measure
180°.
In order to determine if this statement is a
theorem in Hyperbolic Geometry, try to construct a
counter example - an example where the sum DOES NOT
equal 180°.
To do this, first construct three or four random
pairs of intersecting lines. Next, plot a point at the
intersection of each pair of lines by selecting the
"Plot Intersection Point" command from the
"Constructions" menu. Then use the "Measure
Angle" command from the "Measurements" menu
to measure the two angles in each pair of adjacent
angles.
If you succeed in constructing a counter example,
then you have proved the Euclidean Theorem about
adjacent angles false in Hyperbolic Geometry. Failing
to construct a counter example does not prove
anything; however, if, after many tries, you fail to
construct a counter example, then the statement is
likely to be true in Hyperbolic Geometry. .
Note: be aware of rounding. When the
"Measure Angle" command reports that an angle is
45.5° it might actually be 45.4817331°.
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Angles
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WHAT IS A HYPERBOLIC
ANGLE:
Hyperbolic Angles are formed by the intersection
of Hyperbolic rays analogous to the formation of
angles in Euclidean Geometry. The measure
of a Hyperbolic angle, <BAC
is defined to be the measure of the Euclidean
angle, <B'AC',
formed by the Euclidean tangent lines, AB' and
AC'.
MEASURING AN ANGLE
IN NONEUCLID:
You can measure a Hyperbolic angle by selecting
the "Measure Angle" option from the
"Measurement" Menu.
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ACTIVITY: Is
the following theorem true in Hyperbolic
Geometry?
Vertical angles are congruent.
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Parallel
Lines
DEFINITION:
Parallel Lines are lines that do not
intersect.
ACTIVITY:
- Given a point P and any line
 ,
find two lines through P parallel to .
- Can you find more than two lines?
- Is there a "limiting line(s) such that any line
between that "limiting" line(s) and the given line
would intersect ?
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General
Triangles
DEFINITION: A
Triangle is a closed figure formed by three line
segments.
ACTIVITY:
The following is a list of theorems about Triangles in
Euclidean Geometry. Which (if any) are theorems in
Hyperbolic Geometry?
- The sum of the angles of a Triangle is 180
degrees.
- The longest side of a Triangle is opposite the
greatest angle.
- All three altitudes of a Triangle intersect in a
single point. (Hint: To construct an altitude
of a triangle, use the "Draw Perpendicular"
command from the "Constructions" menu.
Click the mouse on any two vertices to define the
base. Then click on the third vertex to draw the
altitude.)
- In a Triangle, the sum of any two sides is always
greater than the length of the third side.
- In a Triangle, if one of the sides is extended,
the exterior angle is greater than either of the
opposite interior angles.
- In a Triangle, the product "base times height" is
the same regardless of which side is chosen as the
base. For example, in triangle ABC, (AB) x (the height
to C) = (BC) x (the height to A).
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Isosceles
Triangles
DEFINITION: An
Isosceles Triangle is a Triangle that has two sides of
the same length.
ACTIVITY: The
following is a list of theorems about Isosceles Triangles
in Euclidean Geometry. Which (if any) are theorems in
Hyperbolic Geometry?
- It is possible to construct an Isosceles Triangle.
(Hint: to prove this statement is true, you must
construct a figure that fits the definition -- a
triangle that has two sides of the same length.)
- The base angles of an Isosceles Triangle are
congruent.
- The altitude of an Isosceles Triangle bisects the
vertex angle and the base.
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Equilateral
Triangles
DEFINITION: An
Equilateral Triangle is a triangle that has three sides
of equal length.
ACTIVITY:
The following is a list of theorems about Equilateral
Triangles in Euclidean Geometry. Which (if any) are
theorems in Hyperbolic Geometry?
- It is possible to construct an Equilateral
Triangle.
- An Equilateral Triangle is also Equiangular (all
three angles have equal measure).
- Each angle of an Equilateral Triangle measures 60
degrees.
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Right
Triangles
DEFINITION: A
Right Triangle is a Triangle that has one right
angle.
ACTIVITY: The
following is a list of theorems about Right Triangles in
Euclidean Geometry. Which (if any) are theorems in
Hyperbolic Geometry?
- It is possible to construct a Right Triangle.
- The Pythagorean Theorem -- In any Right Triangle,
the square of the length of the hypotenuse equals the
sum of the squares of the lengths of the legs.
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Congruent
Triangles
DEFINITION: Two
triangles are Congruent if there exists a correspondence
between them such that the three pairs corresponding
sides hare the same length, and the three pairs
corresponding angles have the same measure.
ACTIVITY:
- A good way to explore properties of Congruent
Triangles is to:
- Construct a triangle.
- Construct an infinite line.
- Use the Reflect command from the Constructions
menu to reflect each of the sides of your triangle
across the infinite line. This
reflected triangle will be congruent to your
original triangle; however, since it is in a
different part of the Hyperbolic plane , it will
appear to have different curvature.
- Use the Move command from the Edit menu to move
the vertex points of the original triangle you
constructed in step (a) or move the Infinite line
you constructed in step (b). The reflection
of the triangle will also move.
- Try making a reflection of a reflection of a
reflection of a reflection. Move the original
vertexes and watch a cool animation of congruent
triangles.
- SSS, and SAS are both theorems in Euclidean
Geometry, are they theorems in Hyperbolic
Geometry?
- In Euclidean Geometry, neither ASS nor AAA, is
sufficient to prove a pair of triangles
congruent. Is either ASS or AAA sufficient to
prove triangles congruent in Hyperbolic Geometry?
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Rectangles
and Squares
DEFINITION: A
Quadrilateral is a closed figure formed by four line
segments. More formally: Given four points A, B, C,
and D, such that they all lie in the same plane, but no
three are collinear. If the segments AB, BC, CD,
and DA intersect only at their end points, then their
union is called a Quadrilateral.
DEFINITION: A
Rectangle is a quadrilateral with four 90°
angles.
DEFINITION: A
Square is a Rectangle with four sides of equal
length.
DEFINITION: A
Regular Quadrilateral is a quadrilateral in which all of
the angles have equal measure and all of the sides have
equal length.
ACTIVITY:
- In Hyperbolic Geometry, rectangles do not exist,
and, therefore, neither do squares. In Hyperbolic
Geometry, if a quadrilateral has 3 right angles, then
the forth angle must be acute. Construct an example of
this.
ACTIVITY: The
following is a list of theorems in Euclidean Geometry.
Which (if any) are theorems in Hyperbolic Geometry?
- It is possible to construct a Regular
Quadrilateral.
- All regular quadrilaterals have four right
angles.
- The two lines passing through the midpoints of the
opposite sides of a regular quadrilateral divide the
regular quadrilateral into four smaller regular
quadrilaterals.
- The diagonals of a regular quadrilateral bisect
each other.
- The diagonals of a regular quadrilateral are
perpendicular.
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Parallelograms
DEFINITION: A
Parallelogram is a quadrilateral in which the opposite
sides are parallel.
ACTIVITY: The
following is a list of theorems about Parallelograms in
Euclidean Geometry. Which (if any) are theorems in
Hyperbolic Geometry?
- It is possible to construct a Parallelogram.
- The opposite sides of a Parallelogram have equal
length.
- The opposite angles of a Parallelogram have equal
measure.
- The diagonals of a Parallelogram bisect each
other.
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Rhombus
DEFINITION: A
Rhombus is a quadrilateral in which all four sides have
equal length.
ACTIVITY: The
following is a list of theorems about Rhombi in Euclidean
Geometry. Which (if any) are theorems in Hyperbolic
Geometry?
- It is possible to construct a Rhombus.
- The opposite angles of a Rhombus are
congruent.
- The diagonals of a Rhombus bisect each other.
- The diagonals of a Rhombus are perpendicular.
- The diagonals of a Rhombus bisect the Rhombus's
angles.
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Polygon
DEFINITION:
A Polygon is a
closed figure formed by three or more line segments.
DEFINITION: A
Regular Polygon is a Polygon in which all of the angles
have equal measure, and all of the sides have equal
length.
ACTIVITY:
- Which of the following Regular Polygons can you
construct with NonEuclid?
- regular triangle
- regular quadrilateral (4 sides)
- regular pentagon (5 sides)
- regular hexagon (6 sides)
- regular heptagon (7 sides)
- regular octagon (8 sides)
- regular nonagon (9 sides)
- regular decagon (10 sides)
- regular dodecagon (12 sides)
- In Euclidean Geometry, any Polygon can be
completely enclosed in some sufficiently large
triangle. This is so obvious a statement that I
have never even seen it written as a theorem.
In, Hyperbolic Geometry, this is not an obvious
statement. Is it a true statement?
- In Euclidean Geometry, any Regular Polygon can be
inscribed in a circle. Is that true in
Hyperbolic Geometry?
- In Euclidean Geometry, any Regular Polygon can be
circumscribed in a circle. Is that true in
Hyperbolic Geometry?
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Circle
DEFINITION: A
Circle is the set of points equal distant from a given
point (the center).
Notice that "having a round shape" is not part of the
definition of a Circle. Personally, I find it very
interesting that circles happen to appear round in both
Euclidean and Hyperbolic geometry.
ACTIVITY:
- In Hyperbolic Geometry, construct a Circle and 8
radii of the Circle.
- In Euclidean Geometry, through any three non
collinear points there passes a circle. Is this
a theorem in Hyperbolic Geometry? (Hint: When looking
for a counter example, recall the Euclidean
construction for circumscribing a triangle.)
- In a Euclidean Geometry Circle, the ratio of
Circumference/Diameter = pi. In Hyperbolic
Geometry, is this ratio a constant for all circles,
and if so is that constant equal to
3.141592654.... Hint: Just as in
Euclidean Geometry, in Hyperbolic Geometry, the
circumference of a circle can be found by the limit of
the perimeter of a series of inscribed regular
polygons. As the number of sides of the regular
polygon increases, the polygon's perimeter becomes a
closer and closer approximation to the circle's
circumference.
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Tessellations
of The Plane
A Tessellation is a covering of an infinite
geometric plane without gaps or overlaps by congruent
figures of one type or a few types.
In Euclidean Geometry, a square can be used to
tessellate the plane; circles, however, will not
tessellate the plane.
The Dutch artist, M.C. Escher
[1902-1972], created many beautiful
tessellations. The figure non the left is a low
resolution copy of Escher's woodcut titled "Geckos"
which is a tessellation of the Euclidean Plane.
Escher also worked extensively with non-Euclidean
geometeries. In particular, his "Circle Limit"
series are all tessellations of the Hyperbolic
plane. The figure on the right, titled "Heaven
and Hell" is one of the works of the "Circle Limit"
series. The demons and angles are each inscribed
in congruent Hyperbolic triangles. The full size
print is really quite attractive.
 
Tessellations by M.C.
Escher
Can you create a tessellate the Hyperbolic
Plane?
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