Mathematics in Kaleidoscopes

Jennifer Li

Department of Mathematics
University of Massachusetts
Amherst, MA
A Tessellation

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Spaces of constant curvature

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Spaces of constant curvature

There are three d-dimensional spaces of constant curvature (d ≥ 2):

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Spaces of constant curvature

There are three d-dimensional spaces of constant curvature (d ≥ 2):

1. Euclidean space Ed

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Spaces of constant curvature

There are three d-dimensional spaces of constant curvature (d ≥ 2):

1. Euclidean space Ed
2. The d-dimensional sphere Sd

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Spaces of constant curvature

There are three d-dimensional spaces of constant curvature (d ≥ 2):

1. Euclidean space Ed
2. The d-dimensional sphere Sd
3. Hyperbolic (or Lobachevskian) space Ld

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Spaces of constant curvature

There are three d-dimensional spaces of constant curvature (d ≥ 2):

1. Euclidean space Ed
2. The d-dimensional sphere Sd
3. Hyperbolic (or Lobachevskian) space Ld
Our focus: dimension d = 2

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Spaces of constant curvature

There are three d-dimensional spaces of constant curvature (d ≥ 2):

1. Euclidean space Ed
2. The d-dimensional sphere Sd
3. Hyperbolic (or Lobachevskian) space Ld
Our focus: dimension d = 2

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Reflection in E2

Reflection across one mirror in E2 (think of R2 ):

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Reflection in E2

Reflection across one mirror in E2 (think of R2 ):

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Reflection in E2

Reflection across one mirror in E2 (think of R2 ):

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Reflection in E2

Reflection across one mirror in E2 (think of R2 ):

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Reflection in E2
Reflection across two mirrors A and B in E2 :

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Reflection in E2
Reflection across two mirrors A and B in E2 :

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Reflection in E2
Reflection across two mirrors A and B in E2 :

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Reflection in E2
Reflection across two mirrors A and B in E2 :

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Reflection in E2
Reflection across two mirrors A and B in E2 :

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This is how a kaleidoscope works!

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Reflection in E2
Reflection across two mirrors A and B in E2 :

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Groups

G = {1, a, b, ab, ba, aba, bab, abab}

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Groups

G = {1, a, b, ab, ba, aba, bab, abab}

Elements of G are motions of the plane E2 .

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Groups

G = {1, a, b, ab, ba, aba, bab, abab}

Elements of G are motions of the plane E2 .

“motion”: one or more reflections across the two mirrors.

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Groups

G = {1, a, b, ab, ba, aba, bab, abab}

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Groups

G = {1, a, b, ab, ba, aba, bab, abab}

The set G is an example of a group. This means:

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Groups

G = {1, a, b, ab, ba, aba, bab, abab}

The set G is an example of a group. This means:

 Any two motions from G results in another motion from G.

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Groups

G = {1, a, b, ab, ba, aba, bab, abab}

The set G is an example of a group. This means:

 Any two motions from G results in another motion from G.

 There is an “neutral” motion in G, denoted by 1, which does not
change anything.

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Groups

G = {1, a, b, ab, ba, aba, bab, abab}

The set G is an example of a group. This means:

 Any two motions from G results in another motion from G.

 There is an “neutral” motion in G, denoted by 1, which does not
change anything.
 Each motion can be “undone” by an inverse motion.

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Groups

G = {1, a, b, ab, ba, aba, bab, abab}

The set G is an example of a group. This means:

 Any two motions from G results in another motion from G.

 There is an “neutral” motion in G, denoted by 1, which does not
change anything.
 Each motion can be “undone” by an inverse motion.
Notice: G has eight elements and they have relations:

a2 = b2 = 1 and (abab)2 = (ab)4 = 1.

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Dihedral Groups

The group G from our example is called a (finite) reflection group.

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Dihedral Groups

The group G from our example is called a (finite) reflection group.

G has a special name: it is called a dihedral group.

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Fundamental domain
Back to our kaleidoscope:

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Fundamental domain
Back to our kaleidoscope:
There are eight regions, called fundamental domains.

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Fundamental domain
Back to our kaleidoscope:
There are eight regions, called fundamental domains.

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Fundamental domain
Back to our kaleidoscope:
There are eight regions, called fundamental domains.

Notice: the fundamental domains

 do not intersect, except at boundaries
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Fundamental domain
Back to our kaleidoscope:
There are eight regions, called fundamental domains.

Notice: the fundamental domains

 do not intersect, except at boundaries
 union to give all of E2
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Fundamental domains

Examples.

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Fundamental domains

Examples.

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Fundamental domains

Examples.

Notice: These fundamental domains have infinite areas

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Fundamental polygons

Example. Let F be an equilateral triangle in E2 .

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Fundamental polygons

Example. Let F be an equilateral triangle in E2 .

Suppose each side of F is a mirror.

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Fundamental polygons
Reflection of F across each mirror results in the following:

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Fundamental polygons
Reflection of F across each mirror results in the following:

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Fundamental polygons
Reflection of F across each mirror results in the following:

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Fundamental polygons
Reflection of F across each mirror results in the following:

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Tessellation!

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Coxeter Polygons

A convex polygon F is called a Coxeter polygon if

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Coxeter Polygons

A convex polygon F is called a Coxeter polygon if

 its vertex angles are of degree π/k for any k = 2, 3, 4, . . . , and

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Coxeter Polygons

A convex polygon F is called a Coxeter polygon if

 its vertex angles are of degree π/k for any k = 2, 3, 4, . . . , and
 the regions formed by reflections across the sides of F do not overlap.

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Coxeter Polygons

A convex polygon F is called a Coxeter polygon if

 its vertex angles are of degree π/k for any k = 2, 3, 4, . . . , and
 the regions formed by reflections across the sides of F do not overlap.
Any Coxeter polygon is a fundamental domain of a reflection group

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Coxeter Polygons

A convex polygon F is called a Coxeter polygon if

 its vertex angles are of degree π/k for any k = 2, 3, 4, . . . , and
 the regions formed by reflections across the sides of F do not overlap.
Any Coxeter polygon is a fundamental domain of a reflection group

Fact. The only Coxeter polygons in E2 are the following:

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Coxeter Polygons

A convex polygon F is called a Coxeter polygon if

 its vertex angles are of degree π/k for any k = 2, 3, 4, . . . , and
 the regions formed by reflections across the sides of F do not overlap.
Any Coxeter polygon is a fundamental domain of a reflection group

Fact. The only Coxeter polygons in E2 are the following:

(i) rectangles

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Coxeter Polygons

A convex polygon F is called a Coxeter polygon if

 its vertex angles are of degree π/k for any k = 2, 3, 4, . . . , and
 the regions formed by reflections across the sides of F do not overlap.
Any Coxeter polygon is a fundamental domain of a reflection group

Fact. The only Coxeter polygons in E2 are the following:

(i) rectangles
(ii) the equilateral triangle

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Coxeter Polygons

A convex polygon F is called a Coxeter polygon if

 its vertex angles are of degree π/k for any k = 2, 3, 4, . . . , and
 the regions formed by reflections across the sides of F do not overlap.
Any Coxeter polygon is a fundamental domain of a reflection group

Fact. The only Coxeter polygons in E2 are the following:

(i) rectangles
(ii) the equilateral triangle
(iii) the isosceles right triangle

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Coxeter Polygons

A convex polygon F is called a Coxeter polygon if

 its vertex angles are of degree π/k for any k = 2, 3, 4, . . . , and
 the regions formed by reflections across the sides of F do not overlap.
Any Coxeter polygon is a fundamental domain of a reflection group

Fact. The only Coxeter polygons in E2 are the following:

(i) rectangles
(ii) the equilateral triangle
(iii) the isosceles right triangle
(iv) the right triangle with angles π/3 and π/6

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Coxeter polygons in the Euclidean plane

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Coxeter polygons in the Euclidean plane

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Coxeter Polygons

Fact. The only Coxeter polygons in E2 are the following:

(i) rectangles
(ii) the equilateral triangle
(iii) the isosceles right triangle
(iv) the right triangle with angles π/3 and π/6

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Coxeter Polygons

Fact. The only Coxeter polygons in E2 are the following:

(i) rectangles
(ii) the equilateral triangle
(iii) the isosceles right triangle
(iv) the right triangle with angles π/3 and π/6

Why?

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Gauss-Bonnet Theorem

Gauss-Bonnet Theorem.

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Gauss-Bonnet Theorem

Gauss-Bonnet Theorem.
Xd : a d-dimensional space of constant curvature 

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Gauss-Bonnet Theorem

Gauss-Bonnet Theorem.
Xd : a d-dimensional space of constant curvature 
P : a polygon in Xd

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Gauss-Bonnet Theorem

Gauss-Bonnet Theorem.
Xd : a d-dimensional space of constant curvature 
P : a polygon in Xd
αi : interior angles of P

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Gauss-Bonnet Theorem

Gauss-Bonnet Theorem.
Xd : a d-dimensional space of constant curvature 
P : a polygon in Xd
αi : interior angles of P
Xn
 · Area(P ) + (π − αi ) = 2π
i=1

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Gauss-Bonnet Theorem

Gauss-Bonnet Theorem.
Xd : a d-dimensional space of constant curvature 
P : a polygon in Xd
αi : interior angles of P
Xn
 · Area(P ) + (π − αi ) = 2π
i=1

Our focus: d = 2

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Gauss-Bonnet Theorem

Our focus: d = 2
Fact. The sum of the angles inside of an n-sided convex polygon P
equals π(n − 2).

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Gauss-Bonnet Theorem

Our focus: d = 2
Fact. The sum of the angles inside of an n-sided convex polygon P
equals π(n − 2).

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Coxeter polygons in the Euclidean plane

F : Coxeter polygon with n sides.

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Coxeter polygons in the Euclidean plane

F : Coxeter polygon with n sides.

Gauss-Bonnet:
Xm
 · Area(P ) + (π − αi ) = 2π
i=1

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Coxeter polygons in the Euclidean plane

F : Coxeter polygon with n sides.

Gauss-Bonnet:
Xm
 · Area(P ) + (π − αi ) = 2π
i=1

Ed : constant curvature  = 0.

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Coxeter polygons in the Euclidean plane

F : Coxeter polygon with n sides.

Gauss-Bonnet:
Xm
 · Area(P ) + (π − αi ) = 2π
i=1

Ed : constant curvature  = 0.
Then the sum of the angles is
Xm
αi = π(n − 2)
i=1

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Coxeter polygons in the Euclidean plane

F : Coxeter polygon with n sides.

Gauss-Bonnet:
Xm
 · Area(P ) + (π − αi ) = 2π
i=1

Ed : constant curvature  = 0.
Then the sum of the angles is
Xm
αi = π(n − 2)
i=1

Average value of the angles:

π(n − 2) 2

=π 1−
n n

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Coxeter polygons in the Euclidean plane

If Coxeter polygon F has n > 4 sides:

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Coxeter polygons in the Euclidean plane

If Coxeter polygon F has n > 4 sides:

2 2
<
n 4

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Coxeter polygons in the Euclidean plane

If Coxeter polygon F has n > 4 sides:

2 2
<
n 4
2 2
1− >1−
n 4

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Coxeter polygons in the Euclidean plane

If Coxeter polygon F has n > 4 sides:

2 2
<
n 4
2 2
1− >1−
n 4
2
2
π
Then π 1 − >π 1− = .
n 4 2

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Coxeter polygons in the Euclidean plane

If Coxeter polygon F has n > 4 sides:

2 2
<
n 4
2 2
1− >1−
n 4
2
2
π
Then π 1 − >π 1− = .
n 4 2
⇒ F has an obtuse angle

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Coxeter polygons in the Euclidean plane

If Coxeter polygon F has n > 4 sides:

2 2
<
n 4
2 2
1− >1−
n 4
2
2
π
Then π 1 − >π 1− = .
n 4 2
⇒ F has an obtuse angle

But this contradicts definition of Coxeter polygon!

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Coxeter polygons in the Euclidean plane

If Coxeter polygon F has n > 4 sides:

2 2
<
n 4
2 2
1− >1−
n 4
2
2
π
Then π 1 − >π 1− = .
n 4 2
⇒ F has an obtuse angle

But this contradicts definition of Coxeter polygon!

Thus n ≤ 4.

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Coxeter polygons in the Euclidean plane

If F has n = 4 sides:

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Coxeter polygons in the Euclidean plane

If F has n = 4 sides:
2
π
π 1− = .
4 2
⇒ F is a rectangle.

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Coxeter polygons in the Euclidean plane

If F has n = 3 sides, then F is a triangle.

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Coxeter polygons in the Euclidean plane

If F has n = 3 sides, then F is a triangle.

π π π
Suppose interior angles of F are , , .
k l m

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Coxeter polygons in the Euclidean plane

If F has n = 3 sides, then F is a triangle.

π π π
Suppose interior angles of F are , , .
k l m
π π π
+ + =π
k l m

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Coxeter polygons in the Euclidean plane

If F has n = 3 sides, then F is a triangle.

π π π
Suppose interior angles of F are , , .
k l m
π π π
+ + =π
k l m
1 1 1
⇒ + + =1
k l m

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Coxeter polygons in the Euclidean plane

If F has n = 3 sides, then F is a triangle.

π π π
Suppose interior angles of F are , , .
k l m
π π π
+ + =π
k l m
1 1 1
⇒ + + =1
k l m
⇒ (k, l, m) = (3, 3, 3), (2, 4, 4), or (2, 3, 6).

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The Coxeter Scheme
A Coxeter scheme is a graph with vertices representing the edge of a
Coxeter polygon, which follows the rules:

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The Coxeter Scheme
A Coxeter scheme is a graph with vertices representing the edge of a
Coxeter polygon, which follows the rules:

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The Coxeter Scheme
A Coxeter scheme is a graph with vertices representing the edge of a
Coxeter polygon, which follows the rules:

These descriptions are useful when working in higher dimensions.

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The Sphere S2

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The Sphere S2

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The Sphere S2

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The Sphere S2
Spherical angles:

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The Sphere S2
Spherical angles:

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The Sphere S2
Spherical angles:

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The Sphere S2
Spherical angles:

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The Sphere S2
Spherical angles:

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The Sphere S2

Gauss-Bonnet:

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The Sphere S2

Gauss-Bonnet:
Xn
 · Area(P ) + (π − αi ) = 2π
i=1

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The Sphere S2

Gauss-Bonnet:
Xn
 · Area(P ) + (π − αi ) = 2π
i=1

Sd : constant curvature  = 1.

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The Sphere S2

Gauss-Bonnet:
Xn
 · Area(P ) + (π − αi ) = 2π
i=1

Sd : constant curvature  = 1.
Xn
⇒ αi = π(n − 2) + Area(P )
i=1

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The Sphere S2

Gauss-Bonnet:
Xn
 · Area(P ) + (π − αi ) = 2π
i=1

Sd : constant curvature  = 1.
Xn
⇒ αi = π(n − 2) + Area(P )
i=1

⇒ sum of interior angles of P > π(n − 2)

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The Sphere S2

For P a triangle: n = 3

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The Sphere S2

For P a triangle: n = 3
π
Suppose the interior angles of P are αi = . Then,
ki

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The Sphere S2

For P a triangle: n = 3
π
Suppose the interior angles of P are αi = . Then,
ki
1 1 1
+ + > π(3 − 2)
k1 k2 k3

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The Sphere S2

For P a triangle: n = 3
π
Suppose the interior angles of P are αi = . Then,
ki
1 1 1
+ + > π(3 − 2)
k1 k2 k3
1 1 1
⇒ + + > 1.
k1 k2 k3

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The Sphere S2

For P a triangle: n = 3
π
Suppose the interior angles of P are αi = . Then,
ki
1 1 1
+ + > π(3 − 2)
k1 k2 k3
1 1 1
⇒ + + > 1.
k1 k2 k3

⇒ (k1 , k2 , k3 ) = (2, 3, 3), (2, 3, 4), (2, 3, 5), or (2, 2, k) for any k ≥ 2.

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The Sphere S2

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Relation to Platonic Solids

There are five platonic solids:

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Relation to Platonic Solids

There are five platonic solids:

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Relation to Platonic Solids

There are five platonic solids:

 Convex polyhedron
 All faces are convex regular polygons
 Same number of faces are joined at each vertex
 Faces only intersect at edges

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The Hyperbolic (Lobachevskian) plane L2

A pseudo-Euclidean space V 2,1 is a 3-dimensional vector space over R,

with scalar product

(x, x) = −x21 + x22 + x23

for any x = (x1 , x2 , x3 ).

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The Hyperbolic (Lobachevskian) plane L2

A pseudo-Euclidean space V 2,1 is a 3-dimensional vector space over R,

with scalar product

(x, x) = −x21 + x22 + x23

for any x = (x1 , x2 , x3 ).

Consider set C = {x ∈ V 2,1 |(x, x) < 0}. Then C = C + ∪ C − , union of

two cones:

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The Hyperbolic (Lobachevskian) plane L2

A pseudo-Euclidean space V 2,1 is a 3-dimensional vector space over R,

with scalar product

(x, x) = −x21 + x22 + x23

for any x = (x1 , x2 , x3 ).

Consider set C = {x ∈ V 2,1 |(x, x) < 0}. Then C = C + ∪ C − , union of

two cones:

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The Hyperbolic (Lobachevskian) plane L2

A pseudo-Euclidean space V 2,1 is a 3-dimensional vector space over R,

with scalar product

(x, x) = −x21 + x22 + x23

for any x = (x1 , x2 , x3 ).

Consider set C = {x ∈ V 2,1 |(x, x) < 0}. Then C = C + ∪ C − , union of

two cones:

All rays in C + form hyperbolic plane L2 .

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The Klein model

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The Klein model

Take a cross section (slice) of C + at x1 = 1 and obtain a unit disk.

Jennifer Li (UMass Amherst) Mathematics in Kaleidoscopes 48 / 63

The Klein model

Take a cross section (slice) of C + at x1 = 1 and obtain a unit disk.

 Unit disk
 Boundary of C +: infinite points of L2
 Chord: straight line in unit disk that comes from rays in V 2,1 which
extend from the origin and into C +

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The Klein model

Lines on the Klein model:

Jennifer Li (UMass Amherst) Mathematics in Kaleidoscopes 49 / 63

The Klein model

Lines on the Klein model:

Jennifer Li (UMass Amherst) Mathematics in Kaleidoscopes 49 / 63

The Klein model

Lines on the Klein model:

Boundary of unit circle: the circle at infinity

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Convex polygons in L2

A convex polygon P in L2 is the intersection of lines in L2 .

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Convex polygons in L2

A convex polygon P in L2 is the intersection of lines in L2 .

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Convex polygons in L2

A convex polygon P in L2 is the intersection of lines in L2 .

Klein model: angles are distorted! (circles too)

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The Poincaré disk model

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The Poincaré disk model

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The Poincaré disk model

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The Poincaré disk model

Some properties:

Jennifer Li (UMass Amherst) Mathematics in Kaleidoscopes 53 / 63

The Poincaré disk model

Some properties:
 Ends of arcs and diameters are perpendicular to the boundary of the
unit disk.

Jennifer Li (UMass Amherst) Mathematics in Kaleidoscopes 53 / 63

The Poincaré disk model

Some properties:
 Ends of arcs and diameters are perpendicular to the boundary of the
unit disk.
 If two arcs do not meet, then they are parallel.

Jennifer Li (UMass Amherst) Mathematics in Kaleidoscopes 53 / 63

The Poincaré disk model

Some properties:
 Ends of arcs and diameters are perpendicular to the boundary of the
unit disk.
 If two arcs do not meet, then they are parallel.
 If two arcs that meet orthogonally represent perpendicular lines.

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The Poincaré disk model

Some properties:
 Ends of arcs and diameters are perpendicular to the boundary of the
unit disk.
 If two arcs do not meet, then they are parallel.
 If two arcs that meet orthogonally represent perpendicular lines.
 Objects close to boundary seem smaller, but in H2 they are the same
size.
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Klein and Poincaré models

Relation with Klein model:

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Klein and Poincaré models

Relation with Klein model:

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Klein and Poincaré models

Relation with Klein model:

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Klein and Poincaré models

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The hyperbolic plane L2

Gauss-Bonnet:

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The hyperbolic plane L2

Gauss-Bonnet:
Xn
 · Area(P ) + (π − αi ) = 2π
i=1

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The hyperbolic plane L2

Gauss-Bonnet:
Xn
 · Area(P ) + (π − αi ) = 2π
i=1

For L2 we have  = −1.

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The hyperbolic plane L2

Gauss-Bonnet:
Xn
 · Area(P ) + (π − αi ) = 2π
i=1

For L2 we have  = −1.

Xn
⇒ αi = π(n − 2) − Area(P )
i=1

Jennifer Li (UMass Amherst) Mathematics in Kaleidoscopes 57 / 63

The hyperbolic plane L2

Gauss-Bonnet:
Xn
 · Area(P ) + (π − αi ) = 2π
i=1

For L2 we have  = −1.

Xn
⇒ αi = π(n − 2) − Area(P )
i=1
⇒ sum of interior angles of P < π(n − 2)

Jennifer Li (UMass Amherst) Mathematics in Kaleidoscopes 57 / 63

The hyperbolic plane L2

Gauss-Bonnet:
Xn
 · Area(P ) + (π − αi ) = 2π
i=1

For L2 we have  = −1.

Xn
⇒ αi = π(n − 2) − Area(P )
i=1
⇒ sum of interior angles of P < π(n − 2)
π π π
⇒ + + ··· + < π(n − 2)
k1 k2 kn

Jennifer Li (UMass Amherst) Mathematics in Kaleidoscopes 57 / 63

The hyperbolic plane L2

Gauss-Bonnet:
Xn
 · Area(P ) + (π − αi ) = 2π
i=1

For L2 we have  = −1.

Xn
⇒ αi = π(n − 2) − Area(P )
i=1
⇒ sum of interior angles of P < π(n − 2)
π π π
⇒ + + ··· + < π(n − 2)
k1 k2 kn
1 1 1
⇒ + + ··· + <n−2
k1 k2 kn

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The hyperbolic plane L2

If P is a triangle in L2 , then we need:

1 1 1
+ + <1
k1 k2 k3

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The hyperbolic plane L2

There are infinitely many possibilities!

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The hyperbolic plane L2

There are infinitely many possibilities!

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The hyperbolic plane L2

Other polygons may tessellate L2 :

Jennifer Li (UMass Amherst) Mathematics in Kaleidoscopes 60 / 63

The hyperbolic plane L2

Other polygons may tessellate L2 :

Which regular hyperbolic n-gons tessellate L2 , with k polygons joining

at each vertex? This happens if and only if the following holds:

Jennifer Li (UMass Amherst) Mathematics in Kaleidoscopes 60 / 63

The hyperbolic plane L2

Other polygons may tessellate L2 :

Which regular hyperbolic n-gons tessellate L2 , with k polygons joining

at each vertex? This happens if and only if the following holds:
1 1 1
+ <
n k 2

Jennifer Li (UMass Amherst) Mathematics in Kaleidoscopes 60 / 63

The hyperbolic plane L2

Other polygons may tessellate L2 :

Which regular hyperbolic n-gons tessellate L2 , with k polygons joining

at each vertex? This happens if and only if the following holds:
1 1 1
+ <
n k 2

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The hyperbolic plane L2

More examples:

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The hyperbolic plane L2

More examples:

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Acknowledgements

Thanks to Paul Hacking, Tetsuya Nakamura, and Luca Schaffler!

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References

 Introduction to Geometry by Harold (H.S.M.) Coxeter

 Visual Geometry and Topology by Anatolij Fomenko
 Reflection groups in Algebraic geometry by Igor Dolgachev
 Alice through looking glass after looking glass: the mathematics of
mirrors and kaleidoscopes by Roe Goodman
 Hyperbolic Geometries and Coxeter Groups by Marilee Murray
 Geometries by Alexei Sossinsky
 Software used to create some hyperbolic tessellations (Klein vs.
Poincaré models) is from the website:
https://dmitrybrant.com/2007/01/24/hyperbolic-tessellations
 All photos, platonic solid images, and other computer generated
hyperbolic tessellations are from Wikipedia

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