Ch. 7: Symmetries of 3D Objects!
Which is most symmetric?
A.
B.
C.
D.
E.
F.
…and what does this question mean?
Many of our previous definitions and theorems generalize effortlessly from the 2D to the 3D setting…
DEFINITONS:
What does this mean?
Intuitively, it is a moving/repositioning of all of space that does not compress, expand, or otherwise distort distances.
Imagine that space is completely filled with transparent ice, with an object entombed somewhere within. A rigid motion moves the entire infinite expanse of ice, and it is called a symmetry of the object if it leaves the object unchanged.
DEFINITONS:
What does this mean?
RIGID MOTIONS OF SPACE
Imagine extending this ball pattern indefinitely
up, down, right, and left to fill up all of space.
Each colored arrow represents a symmetry
of the resulting unbounded pattern.
RIGID MOTIONS OF SPACE
This chapter is about the symmetries of
bounded objects. Bounded objects never
have translation symmetries.
Imagine extending this ball pattern indefinitely
up, down, right, and left to fill up all of space.
Each colored arrow represents a symmetry
of the resulting unbounded pattern.
RIGID MOTIONS OF SPACE
(2) Rotations: A rotation is specified by an axis (line) and an angle.
Three rotation symmetries of the cube
RIGID MOTIONS OF SPACE
Order 4 axis
Order 3 axis
Order 2 axis
DEFINITION: The order of an axis is the order of the smallest non-identity rotation symmetry
about that axis.
(2) Rotations: A rotation is specified by an axis (line) and an angle.
The # times it must be composed with itself to get the identity
RIGID MOTIONS OF SPACE
(3) Reflections: A reflection is specified by a plane.
Which is a plane of reflection symmetry
of the human figure?
Image by YassineMrabet, Wikipedia.org
Think of the plane as a mirror, and imagine each point of space (each speck of ice) moving to the position of its mirror image on the opposite side of the mirror.
RIGID MOTIONS OF SPACE
(3) Reflections: A reflection is specified by a plane.
Which is a plane of reflection symmetry
of the human figure?
Image by YassineMrabet, Wikipedia.org
Think of the plane as a mirror, and imagine each point of space (each speck of ice) moving to the position of its mirror image on the opposite side of the mirror.
The red plane is!
(but not the blue or green)
RIGID MOTIONS OF SPACE
(3) Reflections: A reflection is specified by a plane.
Which is a plane of reflection symmetry
of the human figure?
Image by YassineMrabet, Wikipedia.org
Think of the plane as a mirror, and imagine each point of space (each speck of ice) moving to the position of its mirror image on the opposite side of the mirror.
(The reflected image of a right hand
looks like a left hand. Thus, a reflection
can NOT be physically done to a solid object!
RIGID MOTIONS OF SPACE
(3) Reflections: A reflection is specified by a plane.
Image by YassineMrabet, Wikipedia.org
Think of the plane as a mirror, and imagine each point of space (each speck of ice) moving to the position of its mirror image on the opposite side of the mirror.
DEFINITION: A rigid motion of space is called proper if, after being applied, a solid right hand is still a right hand (or improper if it turns a solid right hand into a left hand).
RIGID MOTIONS OF SPACE
(3) Reflections: A reflection is specified by a plane.
Image by YassineMrabet, Wikipedia.org
Think of the plane as a mirror, and imagine each point of space (each speck of ice) moving to the position of its mirror image on the opposite side of the mirror.
Rotations and translations are proper. They can be physically preformed to a solid object.
A reflection is improper. It can NOT be physically performed to a solid object.
DEFINITION: A rigid motion of space is called proper if, after being applied, a solid right hand is still a right hand (or improper if it turns a solid right hand into a left hand).
RIGID MOTIONS OF SPACE
(3) Reflections: A reflection is specified by a plane.
Several planes of reflection symmetry of the cube.
A plane of reflection symmetry of the tetrahedron.
THEOREM: The symmetries of a 3D object form a group.
The proper symmetries form a subgroup of it.
Called its proper symmetry group.
Called its symmetry group
(or its full symmetry group).
Types or rigid motions: rotations, translations, reflections,….any others?
Types or rigid motions: rotations, translations, reflections,….any others?
CLASSIFICATION OF RIGID MOTIONS OF SPACE: Any rigid motion of space can be obtained by composing rotations, reflections and translations (no more than one of each kind is needed).
Types or rigid motions: rotations, translations, reflections,….any others?
CLASSIFICATION OF RIGID MOTIONS OF SPACE: Any rigid motion of space can be obtained by composing rotations, reflections and translations (no more than one of each kind is needed).
For symmetries of bounded 3D objects, forget about translations…
Types or rigid motions: rotations, translations, reflections,….any others?
CLASSIFICATION OF RIGID MOTIONS OF SPACE: Any rigid motion of space can be obtained by composing rotations, reflections and translations (no more than one of each kind is needed).
THE 3D CENTER POINT THEOREM: Any bounded 3D object has a “center point” that is fixed by each of its symmetries. Moreover, each proper symmetry of the object is a rotation about an axis through this center point.
For symmetries of bounded 3D objects, forget about translations…
Types or rigid motions: rotations, translations, reflections,….any others?
CLASSIFICATION OF RIGID MOTIONS OF SPACE: Any rigid motion of space can be obtained by composing rotations, reflections and translations (no more than one of each kind is needed).
THE 3D CENTER POINT THEOREM: Any bounded 3D object has a “center point” that is fixed by each of its symmetries. Moreover, each proper symmetry of the object is a rotation about an axis through this center point.
NOTE: in 3D, the improper symmetries of a bounded object are NOT necessarily all reflections.
For symmetries of bounded 3D objects, forget about translations…
MAIN GOAL: Classify all ways in which bounded 3D objects can be symmetric:
“Every bounded 3D object is symmetric in the same way as one of these models…”
MAIN GOAL: Classify all ways in which bounded 3D objects can be symmetric:
“Every bounded 3D object is symmetric in the same way as one of these models…”
Rigidly equivalent (or just properly rigidly equivalent)
MAIN GOAL: Classify all ways in which bounded 3D objects can be symmetric:
“Every bounded 3D object is symmetric in the same way as one of these models…”
Rigidly equivalent (or just properly rigidly equivalent)
DEFINITION: Two 3D objects are called (fully) rigidly equivalent if there exists a rigid motion of space which, when applied to the first object, repositions it so that afterwards, the two objects have exactly the same symmetries.
Two 3D objects are called properly rigidly equivalent if there exists a rigid motion of space which, when applied to the first object, repositions it so that afterwards, the two objects have exactly the same proper symmetries.
Rigidly equivalent objects
MAIN GOAL: Classify all ways in which bounded 3D objects can be symmetric:
“Every bounded 3D object is symmetric in the same way as one of these models…”
Rigidly equivalent (or just properly rigidly equivalent)
DEFINITION: Two 3D objects are called (fully) rigidly equivalent if there exists a rigid motion of space which, when applied to the first object, repositions it so that afterwards, the two objects have exactly the same symmetries.
Two 3D objects are called properly rigidly equivalent if there exists a rigid motion of space which, when applied to the first object, repositions it so that afterwards, the two objects have exactly the same proper symmetries.
Rigidly equivalent objects
MAIN GOAL: Classify all ways in which bounded 3D objects can be symmetric:
“Every bounded 3D object is symmetric in the same way as one of these models…”
Rigidly equivalent (or just properly rigidly equivalent)
DEFINITION: Two 3D objects are called (fully) rigidly equivalent if there exists a rigid motion of space which, when applied to the first object, repositions it so that afterwards, the two objects have exactly the same symmetries.
Two 3D objects are called properly rigidly equivalent if there exists a rigid motion of space which, when applied to the first object, repositions it so that afterwards, the two objects have exactly the same proper symmetries.
THEOREM:
If two 3D objects are rigidly equivalent, then their symmetry groups are isomorphic.
If two 3D objects are properly rigidly equivalent, then their proper symmetry groups are isomorphic.
Essentially two-dimensional objects
Essentially two-dimensional objects
The simplest kind of 3D object: the study of its proper symmetry group
reduces to just studying something two-dimensional.
Essentially two-dimensional objects
The simplest kind of 3D object: the study of its proper symmetry group
reduces to just studying something two-dimensional.
Essentially two-dimensional objects
The simplest kind of 3D object: the study of its proper symmetry group
reduces to just studying something two-dimensional.
Essentially two-dimensional objects
The simplest kind of 3D object: the study of its proper symmetry group
reduces to just studying something two-dimensional.
Essentially two-dimensional objects
The simplest kind of 3D object: the study of its proper symmetry group
reduces to just studying something two-dimensional.
Essentially two-dimensional objects
The simplest kind of 3D object: the study of its proper symmetry group
reduces to just studying something two-dimensional.
Essentially two-dimensional objects
The simplest kind of 3D object: the study of its proper symmetry group
reduces to just studying something two-dimensional.
The isomorphism matches:
Essentially two-dimensional objects
The simplest kind of 3D object: the study of its proper symmetry group
reduces to just studying something two-dimensional.
The isomorphism matches:
The Greek word “dihedral” comes from this 3D viewpoint.
Essentially two-dimensional objects
Essentially two-dimensional objects
Essentially two-dimensional objects
Essentially two-dimensional objects
THEOREM: Given any essentially two-dimensional object with finitely many symmetries, one of the following is true:
(1) It has no rotation axes, in which case it is asymmetric or has bilateral symmetry.
(2) It has exactly one rotation axis, in which case it is properly rigidly equivalent to a beveled regular polygon.
(3) It has one main rotation axis plus additional flip axes, in which case it is properly rigidly equivalent to a thick regular polygon.
Essentially two-dimensional objects
THEOREM: Given any essentially two-dimensional object with finitely many symmetries, one of the following is true:
(1) It has no rotation axes, in which case its only symmetries are the identity and possibly also an improper symmetry.
(2) It has exactly one rotation axis, in which case it is properly rigidly equivalent to a beveled regular polygon.
(3) It has one main rotation axis plus additional flip axes, in which case it is properly rigidly equivalent to a thick regular polygon.
Essentially two-dimensional objects
THEOREM: Given any essentially two-dimensional object with finitely many symmetries, one of the following is true:
(1) It has no rotation axes, in which case its only symmetries are the identity and possibly also an improper symmetry.
(2) It has exactly one rotation axis, in which case it is properly rigidly equivalent to a beveled regular polygon.
(3) It has one main rotation axis plus additional flip axes, in which case it is properly rigidly equivalent to a thick regular polygon.
Essentially two-dimensional objects
THEOREM: Given any essentially two-dimensional object with finitely many symmetries, one of the following is true:
(1) It has no rotation axes, in which case its only symmetries are the identity and possibly also an improper symmetry.
(2) It has exactly one rotation axis, in which case it is properly rigidly equivalent to a beveled regular polygon.
(3) It has one main rotation axis plus additional flip axes, in which case it is properly rigidly equivalent to a thick regular polygon.
Essentially two-dimensional objects
THEOREM: Given any essentially two-dimensional object with finitely many symmetries, one of the following is true:
(1) It has no rotation axes, in which case its only symmetries are the identity and possibly also an improper symmetry.
(2) It has exactly one rotation axis, in which case it is properly rigidly equivalent to a beveled regular polygon.
(3) It has one main rotation axis plus additional flip axes, in which case it is properly rigidly equivalent to a thick regular polygon.
Essentially two-dimensional objects
What about 3D objects that are NOT essentially two-dimensional?
Essentially two-dimensional objects
What about 3D objects that are NOT essentially two-dimensional?
Such an object might have lots of high-order axes.
Essentially two-dimensional objects
What about 3D objects that are NOT essentially two-dimensional?
Such an object might have lots of high-order axes.
Fortunately, it will be enough to understand just THREE such objects…
Tetrahedron Cube Dodecahedron
Tetrahedron Cube Dodecahedron
How many proper (rotation) symmetries does each object have?
Tetrahedron Cube Dodecahedron
| F | S |
tetrahedron | 4 | 3 |
cube | 6 | 4 |
dodecahedron | 12 | 5 |
S = # sides to a face
F = # faces
A proper symmetry is specified by: (1) which face goes down?
(2) how is this bottom face rotated?
Tetrahedron Cube Dodecahedron
| F | S | # proper symmetries |
tetrahedron | 4 | 3 | 12 |
cube | 6 | 4 | 24 |
dodecahedron | 12 | 5 | 60 |
F = # faces
A proper symmetry is specified by: (1) which face goes down?
(2) how is this bottom face rotated?
S = # sides to a face
_____ axes of order 3. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 2. TOTAL = _____ rotations.
(not counting Identity)
TOTAL PROPER SYMETRIES = _______
(Remember to count the identity)
TOTAL SYMETRIES = _______
Rotation axes of the tetrahedron.
Rotation axes of the tetrahedron.
_____ axes of order 3. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 2. TOTAL = _____ rotations.
(not counting Identity)
TOTAL PROPER SYMETRIES = _______
8
4
(Remember to count the identity)
TOTAL SYMETRIES = _______
Rotation axes of the tetrahedron.
_____ axes of order 3. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 2. TOTAL = _____ rotations.
(not counting Identity)
TOTAL PROPER SYMETRIES = _______
3
8
3
4
12
(Remember to count the identity)
TOTAL SYMETRIES = _______
Rotation axes of the tetrahedron.
_____ axes of order 3. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 2. TOTAL = _____ rotations.
(not counting Identity)
TOTAL PROPER SYMETRIES = _______
3
8
3
4
12
(Remember to count the identity)
TOTAL SYMETRIES = _______
24
The Zero-or-Equal Theorem
is valid for 3D objects!
The symmetry group of the tetrahedron.
Label the vertices A,B,C,D. This is the starting position.
The symmetry group of the tetrahedron.
Match each symmetry with the way it permutes the vertices; that is, the resulting “word” read in this order: top, right, left, front.
Label the vertices A,B,C,D. This is the starting position.
The symmetry group of the tetrahedron.
Match each symmetry with the way it permutes the vertices; that is, the resulting “word” read in this order: top, right, left, front.
= “swap A & B”
The symmetry group of the tetrahedron.
The symmetry group of the tetrahedron.
The symmetry group of the tetrahedron.
Rotations permute the
Vertices in even ways
2 swaps
One cycle of length 3
The rotation axes of a cube
_____ axes of order 4. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 3. TOTAL = _____ rotations.
(not counting Identity)
TOTAL PROPER SYMETRIES = _______
(Remember to count the identity)
TOTAL SYMETRIES = _______
_____ axes of order 2. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 4. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 3. TOTAL = _____ rotations.
(not counting Identity)
TOTAL PROPER SYMETRIES = _______
(Remember to count the identity)
TOTAL SYMETRIES = _______
_____ axes of order 2. TOTAL = _____ rotations.
(not counting Identity)
3
9
The rotation axes of a cube
_____ axes of order 4. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 3. TOTAL = _____ rotations.
(not counting Identity)
TOTAL PROPER SYMETRIES = _______
(Remember to count the identity)
TOTAL SYMETRIES = _______
_____ axes of order 2. TOTAL = _____ rotations.
(not counting Identity)
3
9
4
8
The rotation axes of a cube
_____ axes of order 4. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 3. TOTAL = _____ rotations.
(not counting Identity)
TOTAL PROPER SYMETRIES = _______
(Remember to count the identity)
TOTAL SYMETRIES = _______
_____ axes of order 2. TOTAL = _____ rotations.
(not counting Identity)
3
9
4
8
6
6
The rotation axes of a cube
_____ axes of order 4. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 3. TOTAL = _____ rotations.
(not counting Identity)
TOTAL PROPER SYMETRIES = _______
(Remember to count the identity)
TOTAL SYMETRIES = _______
_____ axes of order 2. TOTAL = _____ rotations.
(not counting Identity)
3
9
4
8
6
6
24
48
The rotation axes of a cube
TOTAL PROPER SYMETRIES = _______
TOTAL SYMETRIES = _______
24
48
We might guess the proper symmetry group is a permutation
or alternating group. Only one has the right size:
|P3| = 6 |A3| = 3
|P4| = 24 |A4| = 12
|P5| = 120 |A5| = 60
|P6| = 720 |A6| = 380
The rotation axes of a cube
TOTAL PROPER SYMETRIES = _______
TOTAL SYMETRIES = _______
24
48
CONJECTURE: The proper symmetry group of a cube is isomorphic to P4.
But what does a cube have 4 of which get
permuted by its proper symmetries?
TOTAL PROPER SYMETRIES = _______
TOTAL SYMETRIES = _______
24
48
CONJECTURE: The proper symmetry group of a cube is isomorphic to P4.
But what does a cube have 4 of which get
permuted by its proper symmetries?
It has 4 colored diagonals!
CONJECTURE: The proper symmetry group of a cube is isomorphic to P4.
But what does a cube have 4 of which get
permuted by its proper symmetries?
To prove the conjecture, we must verify that the 24 proper symmetries match 1-to-1 with the 24 four-letter words.
CONJECTURE: The proper symmetry group of a cube is isomorphic to P4.
But what does a cube have 4 of which get
permuted by its proper symmetries?
To prove the conjecture, we must verify that the 24 proper symmetries match 1-to-1 with the 24 four-letter words.
All 24 words are achieved as positions of the cube!
THEOREM: The proper symmetry group of a cube is isomorphic to P4.
But what does a cube have 4 of which get
permuted by its proper symmetries?
To prove the conjecture, we must verify that the 24 proper symmetries match 1-to-1 with the 24 four-letter words.
All 24 words are achieved as positions of the cube!
The rotation axes of a dodecahedron.
_____ axes of order 3. TOTAL = _____ rotations.
(not counting Identity)
TOTAL PROPER SYMETRIES = _______
(Remember to count the identity)
TOTAL SYMETRIES = _______
_____ axes of order 2. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 5. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 3. TOTAL = _____ rotations.
(not counting Identity)
TOTAL PROPER SYMETRIES = _______
(Remember to count the identity)
TOTAL SYMETRIES = _______
_____ axes of order 2. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 5. TOTAL = _____ rotations.
(not counting Identity)
6
24
The rotation axes of a dodecahedron.
_____ axes of order 3. TOTAL = _____ rotations.
(not counting Identity)
TOTAL PROPER SYMETRIES = _______
(Remember to count the identity)
TOTAL SYMETRIES = _______
_____ axes of order 2. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 5. TOTAL = _____ rotations.
(not counting Identity)
6
24
10
20
The rotation axes of a dodecahedron.
_____ axes of order 3. TOTAL = _____ rotations.
(not counting Identity)
TOTAL PROPER SYMETRIES = _______
(Remember to count the identity)
TOTAL SYMETRIES = _______
_____ axes of order 2. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 5. TOTAL = _____ rotations.
(not counting Identity)
6
24
15
15
10
20
The rotation axes of a dodecahedron.
_____ axes of order 3. TOTAL = _____ rotations.
(not counting Identity)
TOTAL PROPER SYMETRIES = _______
(Remember to count the identity)
TOTAL SYMETRIES = _______
_____ axes of order 2. TOTAL = _____ rotations.
(not counting Identity)
_____ axes of order 5. TOTAL = _____ rotations.
(not counting Identity)
6
24
15
15
10
20
60
120
The rotation axes of a dodecahedron.
TOTAL PROPER SYMETRIES = _______
TOTAL SYMETRIES = _______
60
120
We might guess the proper symmetry group is a permutation
or alternating group. Only one has the right size:
|P3| = 6 |A3| = 3
|P4| = 24 |A4| = 12
|P5| = 120 |A5| = 60
|P6| = 720 |A6| = 380
The rotation axes of a dodecahedron.
?
?
TOTAL PROPER SYMETRIES = _______
TOTAL SYMETRIES = _______
60
120
QUESTION: Is the proper symmetry group of a dodecahedron is isomorphic to A5. Is the full symmetry group isomorphic to P5?
What does a dodecahedron have 5 of which get
permuted by its proper symmetries?
The rotation axes of a dodecahedron.
QUESTION: Is the proper symmetry group of a dodecahedron is isomorphic to A5. Is the full symmetry group isomorphic to P5?
What does a dodecahedron have 5 of which get
permuted by its proper symmetries?
The rotation axes of a dodecahedron.
It has 5 colors!
QUESTION: Is the proper symmetry group of a dodecahedron is isomorphic to A5. Is the full symmetry group isomorphic to P5?
What does a dodecahedron have 5 of which get
permuted by its proper symmetries?
The rotation axes of a dodecahedron.
It has 5 colors!
The 5 colors represent
the 5 possible ways to
inscribe a largest-possible cube in the dodecahedron.
QUESTION: Is the proper symmetry group of a dodecahedron is isomorphic to A5. Is the full symmetry group isomorphic to P5?
What does a dodecahedron have 5 of which get
permuted by its proper symmetries?
Do the proper symmetries match 1-to-1 with the
even permutations?
Do the improper symmetries match 1-to-1 with
the odd permutations?
THEOREM: The proper symmetry group of a dodecahedron is isomorphic to A5.
Later we’ll figure out it’s full symmetry group.
HINT: It’s not P5.
Tetrahedron Cube Dodecahedron
A4 (12)
P4 (24)
A5 (60)
Proper symmetry
group (SIZE)
SUMMARY
Tetrahedron Cube Dodecahedron
A4 (12)
P4 (24)
A5 (60)
Proper symmetry
group (SIZE)
SUMMARY
Why are these three objects so important?
Tetrahedron Cube Dodecahedron
A4 (12)
P4 (24)
A5 (60)
Proper symmetry
group (SIZE)
THE CLASSIFICATION THEOREM: Every bounded 3D object with finitely many symmetries is either essentially two-dimensional or is properly rigidly equivalent to a tetrahedron, cube, or dodecahedron.
Tetrahedron Cube Dodecahedron
A4 (12)
P4 (24)
A5 (60)
Proper symmetry
group (SIZE)
How would you distribute 12 spikes symmetrically around
a sphere? 20? 30? 100?
THE CLASSIFICATION THEOREM: Every bounded 3D object with finitely many symmetries is either essentially two-dimensional or is properly rigidly equivalent to a tetrahedron, cube, or dodecahedron.
Tetrahedron Cube Dodecahedron
A4 (12)
P4 (24)
A5 (60)
Proper symmetry
group (SIZE)
How would you distribute 12 spikes symmetrically around
a sphere? 20? 30? 100?
You can’t! More than 60 can NOT be done!!!
THE CLASSIFICATION THEOREM: Every bounded 3D object with finitely many symmetries is either essentially two-dimensional or is properly rigidly equivalent to a tetrahedron, cube, or dodecahedron.
You can’t symmetrically distribute more
than 60 points around a sphere!
“Symmetrically” means here that a proper
symmetry can move any point to any other point.
Chirality
DEFINITION: A 3D object is called chiral if all of its symmetries are proper.
(analogous to the word “oriented” for two-dimensional objects)
Chirality
CHIRAL
NOT CHIRAL
DEFINITION: A 3D object is called chiral if all of its symmetries are proper.
Chirality
All chiral objects look and act differently in Alice’s Looking-glass world.
DEFINITION: A 3D object is called chiral if all of its symmetries are proper.
Chirality
All chiral objects look and act differently in Alice’s Looking-glass world.
DEFINITION: A 3D object is called chiral if all of its symmetries are proper.
Chirality
Chiral tetrahedron, chiral cube and chiral dodecahedron (reflected in a mirror)
If it’s properly rigidly equivalent to a | then it’s rigidly equivalent to a |
…tetrahedron… | …tetrahedron, chiral tetrahedron, or volleyball. |
…cube… | …cube or chiral cube. |
…dodecahedron… | …dodecahedron or chiral dodecahedron. |
If it’s properly rigidly equivalent to a | then it’s rigidly equivalent to a |
…tetrahedron… | …tetrahedron, chiral tetrahedron, or volleyball. |
…cube… | …cube or chiral cube. |
…dodecahedron… | …dodecahedron or chiral dodecahedron. |
If it’s properly rigidly equivalent to a | then it’s rigidly equivalent to a |
…tetrahedron… | …tetrahedron, chiral tetrahedron, or volleyball. |
…cube… | …cube or chiral cube. |
…dodecahedron… | …dodecahedron or chiral dodecahedron. |
The volleyball and tetrahedron are properly rigidly equivalent.
More about the full symmetry groups of bounded 3D objects
The improper rigid motion called central inversion is best described as “reflecting through a point”
Central inversion through the green point
More about the full symmetry groups of bounded 3D objects
The improper rigid motion called central inversion is best described as “reflecting through a point”
Central inversion through the green point
More about the full symmetry groups of bounded 3D objects
But we can’t just make up a new type of rigid motion because they are all just compositions of the familiar types.
The improper rigid motion called central inversion is best described as “reflecting through a point”
Central inversion through the green point
is the same as rotating and then reflecting
More about the full symmetry groups of bounded 3D objects
The improper rigid motion called central inversion is best described as “reflecting through a point”
Central inversion through the green point
is the same as rotating and then reflecting
Central inversion is a symmetry of this outward explosion
More about the full symmetry groups of bounded 3D objects
The improper rigid motion called central inversion is best described as “reflecting through a point”
Central inversion through the green point
is the same as rotating and then reflecting
Central inversion exchanges
these two tetrahedron frames
More about the full symmetry groups of bounded 3D objects
The improper rigid motion called central inversion is best described as “reflecting through a point”
An object is called centrally symmetric if central inversion (across its center point) is a symmetry of the object.
Centrally symmetric
NOT Centrally symmetric
More about the full symmetry groups of bounded 3D objects
The improper rigid motion called central inversion is best described as “reflecting through a point”
An object is called centrally symmetric if central inversion (across its center point) is a symmetry of the object.
Centrally symmetric
More about the full symmetry groups of bounded 3D objects
Centrally symmetric
More about the full symmetry groups of bounded 3D objects
More about the full symmetry groups of bounded 3D objects
More about the full symmetry groups of bounded 3D objects
Central inversion swaps upper and lower case letters.
More about the full symmetry groups of bounded 3D objects
P4 = { ABCD, ABDC, ACBD, ACDB, ADBC, ADCB,
BACD, BADC, BCAD, BCDA, BDAC, BDCA,
CABD, CADB, CBAD, CBDA, CDAB, CDBA,
DABC, DACB, DBAC, DBCA, DCAB, DCBA }
Central inversion swaps upper and lower case letters.
More about the full symmetry groups of bounded 3D objects
P4 = { ABCD, ABDC, ACBD, ACDB, ADBC, ADCB,
BACD, BADC, BCAD, BCDA, BDAC, BDCA,
CABD, CADB, CBAD, CBDA, CDAB, CDBA,
DABC, DACB, DBAC, DBCA, DCAB, DCBA }
( BCDA , 0 ) = the proper symmetry that rotates the cube to the BCDA position
( BCDA , 1 ) = the improper symmetry that rotates the cube to the BCDA position
and then performs central inversion
Central inversion swaps upper and lower case letters.
More about the full symmetry groups of bounded 3D objects
The Full-vs-Proper Theorem even applies to some
objects that are not centrally symmetric…
More about the full symmetry groups of bounded 3D objects
EXERCISE: Identify the full symmetry group of a thick hexagon.
More about the full symmetry groups of bounded 3D objects
EXERCISE: Identify the full symmetry group of a thick hexagon.
The reflection across the plane that slices it like a bagle commutes with all of its proper symmetries – it swaps upper and lower case letters.
More about the full symmetry groups of bounded 3D objects
EXERCISE: Identify the full symmetry group of a thick hexagon.
The reflection across the plane that slices it like a bagle commutes with all of its proper symmetries – it swaps upper and lower case letters.
More about the full symmetry groups of bounded 3D objects
EXERCISE: Identify the full symmetry group of a thick hexagon.
More about the full symmetry groups of bounded 3D objects
EXERCISE: Identify the full symmetry group of a thick hexagon.
More about the full symmetry groups of bounded 3D objects
EXERCISE: Identify the full symmetry group of a thick hexagon.
More about the full symmetry groups of bounded 3D objects
EXERCISE: Identify the full symmetry group of a beveled square.
More about the full symmetry groups of bounded 3D objects
EXERCISE: Identify the full symmetry group of a beveled square.
The Full-vs-Proper Theorem doesn’t apply here.
More about the full symmetry groups of bounded 3D objects
EXERCISE: Identify the full symmetry group of a beveled square.
The Full-vs-Proper Theorem doesn’t apply here.
Reflection planes of the beveled square
Correspond to reflection lines of a 2D square.
| proper sym. gp. (size) | Full sym. gp. (size) |
tetrahedron | A4 (12) | P4 (24) |
chiral tetrahedron | A4 (12) | |
volleyball | A4×C2 (24) | |
cube | P4 (24) | P4×C2 (48) |
chiral cube | P4 (24) | |
dodecahedron | A5 (60) | A5×C2 (120) |
chiral dodecahedron | A5 (60) | |
Beveled n-gon | Cn (n) | Dn (2n) |
Thick n-gon | Dn (2n) | Dn×C2 (4n) |
More about the full symmetry groups of bounded 3D objects
SUMMARY
For each object, decide:
For each object, decide:
For each object, decide:
For each object, decide:
For each object, decide:
For each object, decide:
For each object, decide:
For each object, decide:
For each object, decide: