# Graded Symmetry Groups

Plane and Simple.

the universally accepted unwritten axiom

AGACSE2021

Steven De Keninck (enkimute)

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https://slides.com/d/0A4Ehfc/live

# AGACSE2021

On Lie groups defining automorphisms that leave invariant fundamental subspaces of geometric algebra.


Dmitry Shirokov

# Graded Symmetry Groups

Plane and Simple.

the universally accepted unwritten axiom

AGACSE2021

L’application des mêmes idées de dualité peut s’étendre à la Mécanique. En effet, l’élément primitif des corps auquel on applique d’abord les premiers principes de cette science, est, comme dans la Géométrie ancienne, le point mathématique. Ne sommes-nous pas autorisés à penser, maintenant, qu’en prenant le plan pour l’élément de l’étendue, et non plus le point, on sera conduit à d’autres doctrines, faisant pour ainsi dire une nouvelle science?

- Michel Chasles, 1875

# Graded Symmetry Groups

Plane and Simple.

the universally accepted unwritten axiom

AGACSE2021

The application of the same ideas of duality can be extended to Mechanics. Indeed, the primitive element of bodies to which the first principles of this science are applied is, as in ancient Geometry, the mathematical point. Are we not permitted to think, now, that by taking the plane for the basic element, and no longer the point, we shall be led to other doctrines, making, as it were, a new science?

- Michel Chasles, 1875

# Graded Symmetry Groups

Plane and Simple.

the universally accepted unwritten axiom

AGACSE2021

The application of the same ideas of duality can be extended to Mechanics. Indeed, the primitive element of bodies to which the first principles of this science are applied is, as in ancient Geometry, the mathematical point. Are we not permitted to think, now, that by taking the plane for the basic element, and no longer the point, we shall be led to other doctrines, making, as it were, a new science?

- Michel Chasles, 1875

1. Symmetry Groups

2. Geometry

3. Rigid Body Dynamics

• classic approach
• k-reflections
• geometric gauges
• Mozzi-Chasles
• coordinate agnostic
• dimension agnostic
• metric agnostic
• Forques
• Demo

# Graded Symmetry Groups

Plane and Simple.

the universally accepted unwritten axiom

AGACSE2021

The application of the same ideas of duality can be extended to Mechanics. Indeed, the primitive element of bodies to which the first principles of this science are applied is, as in ancient Geometry, the mathematical point. Are we not permitted to think, now, that by taking the plane for the basic element, and no longer the point, we shall be led to other doctrines, making, as it were, a new science?

- Michel Chasles, 1875

1. Symmetry Groups

2. Geometry

3. Rigid Body Dynamics

• classic approach
• k-reflections
• geometric gauges
• Mozi-Chasles
• coordinate agnostic
• dimension agnostic
• metric agnostic
• Forques
• Demo

All our examples will be using the Euclidean groups in 2 and 3 dimensions.

Why?

• these are non-simple Lie groups
• the natural choice for intuition

# Graded Symmetry Groups

Plane and Simple.

the universally accepted unwritten axiom

AGACSE2021

The application of the same ideas of duality can be extended to Mechanics. Indeed, the primitive element of bodies to which the first principles of this science are applied is, as in ancient Geometry, the mathematical point. Are we not permitted to think, now, that by taking the plane for the basic element, and no longer the point, we shall be led to other doctrines, making, as it were, a new science?

- Michel Chasles, 1875

1. Symmetry Groups

2. Geometry

3. Rigid Body Dynamics

• classic approach
• k-reflections
• geometric gauges
• Mozi-Chasles
• coordinate agnostic
• dimension agnostic
• metric agnostic
• Forques
• Demo

All our examples will be using the Euclidean groups in 2 and 3 dimensions.

Why?

• these are non-simple Lie groups
• the natural choice for intuition

a vector space

transformations on a vector space

\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}
\begin{bmatrix} m_{11} & \dots & m_{1n} \\ \vdots & & \vdots \\ m_{n1} & \dots & m_{nn} \end{bmatrix}

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

4x4 matrix

6x6 matrix

4x4 matrix

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups

a vector space

transformations on a vector space

\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}
\begin{bmatrix} x_{11} & \dots & x_{1n} \\ \vdots & & \vdots \\ x_{n1} & \dots & x_{nn} \end{bmatrix}

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

4x4 matrix

6x6 matrix

4x4 matrix

In this model, perfectly reasonable starting point.

Also valid, but why would we?

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups

a vector space

transformations on a vector space

\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}
\begin{bmatrix} x_{11} & \dots & x_{1n} \\ \vdots & & \vdots \\ x_{n1} & \dots & x_{nn} \end{bmatrix}

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

4x4 matrix

6x6 matrix

4x4 matrix

a graded vector space

x \in \mathbb R_{p,q,r}

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

versors/rotors

R \in \mathbb R_{p,q,r}^+

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups

a vector space

transformations on a vector space

\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}
\begin{bmatrix} x_{11} & \dots & x_{1n} \\ \vdots & & \vdots \\ x_{n1} & \dots & x_{nn} \end{bmatrix}

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

4x4 matrix

6x6 matrix

4x4 matrix

a graded vector space

x \in \mathbb R_{p,q,r}

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

versors/rotors

R \in \mathbb R_{p,q,r}^+

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   k-reflections

a graded vector space

x \in \mathbb R_{p,q,r}

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

versors/rotors

R \in \mathbb R_{p,q,r}^+

start from Hamilton's observation that all isometries can be generated by composing reflections. Then stick with it.

$$E(2)$$ - The Euclidean Group of the plane

# reflections Isometry
0 identity
1 reflection
2 rotation/translation
3 glide reflection
4 ?  → 2

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   k-reflections

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

start from Hamilton's observation that all isometries can be generated by composing reflections. Then stick with it.

$$E(2)$$ - The Euclidean Group of the plane

# reflections Isometry
0 identity
1 reflection
2 rotation/translation
3 glide reflection
4 ?  → 2
Invariant
plane
line
point /  ∞point
line + ∞point
→ 2

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   k-reflections

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

start from Hamilton's observation that all isometries can be generated by composing reflections. Then stick with it.

$$E(2)$$ - The Euclidean Group of the plane

# reflections Isometry
0 identity
1 reflection
2 rotation/translation
3 glide reflection
Invariant
plane
line
point /  ∞point
line + ∞point
GA
scalar
vector
bivector
trivector

Using Chasles' hyperplanes as vectors in a GA gives us exactly the structure we need.

the reflections give us everything.

Geometric Algebra Sandwich

$$-a b a^{-1}$$

Group Theory Conjugation

$$a b a^{-1}$$

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   k-reflections

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

$$E(2)$$ - The Euclidean Group of the plane

# reflections Isometry
0 identity
1 reflection
2 rotation/translation
3 glide reflection
Invariant
plane
line
point /  ∞point
line + ∞point

Geometric Algebra Sandwich

$$-a b a^{-1}$$

GA
scalar
vector
bivector
trivector

Using Chasles' hyperplanes as vectors in a GA gives us exactly the structure we need.

the reflections give us everything.

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   geometric gauges

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   geometric gauges

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   geometric gauges

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   Mozzi-Chasles Theorem

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   Mozzi-Chasles Theorem

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

The Mozzi-Chasles Theorem :

Every rigid body motion can be decomposed into a rotation around followed or preceded by a a translation along a single line.

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   Mozi-Chasles Theorem

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

The Mozzi-Chasles Theorem :

Every rigid body motion can be decomposed into a rotation around followed or preceded by a a translation along a single line.

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   Mozi-Chasles Theorem

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

The Mozzi-Chasles Theorem :

Every quadreflection can be decomposed into a rotation around followed or preceded by a a translation along a single line.

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   Mozi-Chasles Theorem

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

The Mozzi-Chasles Theorem :

Every quadreflection can be decomposed into a commuting rotation and translation

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   Mozi-Chasles Theorem

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

The Mozzi-Chasles Theorem :

Every quadreflection can be decomposed into commuting bireflections

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   Mozi-Chasles Theorem

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

Our final theorem :

Every k-reflection can be decomposed into $$\lceil \frac k 2 \rceil$$ commuting parts.

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   Mozi-Chasles Theorem

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

Our final theorem :

Every k-reflection can be decomposed into $$\lceil \frac k 2 \rceil$$ commuting parts.

any $$k$$, even and odd

any # dimensions,

any metric

closed form exp and log. $$e^{a+b} = e^ae^b$$ iff $$ab = ba$$

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   Mozi-Chasles Theorem

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

Our final version :

Every k-reflection can be decomposed into $$\lceil \frac k 2 \rceil$$ commuting parts.

any $$k$$, even and odd

any # dimensions,

any metric

closed form exp and log. $$e^{a+b} = e^ae^b$$ iff $$ab = ba$$

Martin's talk follows up on this and goes into all the algebraic details!

Today, lets envision how this could make a difference in applications and intuition.

2. Geometry | coordinate, dimension and metric agnostic

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   Mozi-Chasles Theorem

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

Our final version :

Every k-reflection can be decomposed into $$\lceil \frac k 2 \rceil$$ commuting parts.

any $$k$$, even and odd

any # dimensions,

any metric

closed form exp and log. $$e^{a+b} = e^ae^b$$ iff $$ab = ba$$

Martin's talk follows up on this and goes into all the algebraic details!

Today, lets envision how this could make a difference in applications and intuition.

2. Geometry | coordinate, dimension and metric agnostic

this is not a vector.

ceci n'est pas un vecteur

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   Mozi-Chasles Theorem

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

any $$k$$, even and odd

any # dimensions,

any metric

closed form exp and log. $$e^{a+b} = e^ae^b$$ iff $$ab = ba$$

Martin's talk follows up on this and goes into all the algebraic details!

Today, lets envision how this could make a difference in applications and intuition.

2. Geometry | coordinate, dimension and metric agnostic

this is a vector.

ceci n'est pas un vecteur

how many dimensions ?

does it have an orientation ?

# Graded Symmetry Groups

AGACSE2021

1. Symmetry Groups   |   Mozi-Chasles Theorem

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

any $$k$$, even and odd

any # dimensions,

any metric

closed form exp and log. $$e^{a+b} = e^ae^b$$ iff $$ab = ba$$

Martin's talk follows up on this and goes into all the algebraic details!

Today, lets envision how this could make a difference in applications and intuition.

2. Geometry | coordinate, dimension and metric agnostic

this is a vector.

ceci n'est pas un vecteur

$$x^2 = \lVert x \rVert^2$$

only its length is well defined.

# Graded Symmetry Groups

AGACSE2021

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

any $$k$$, even and odd

any # dimensions,

any metric

closed form exp and log. $$e^{a+b} = e^ae^b$$ iff $$ab = ba$$

2. Geometry | coordinate, dimension and metric agnostic

this is a vector.

$$x^2 = \lVert x \rVert^2$$

When a second non colinear vector is added

• we can measure two lengths
• we can measure one angle
• we know our space is >= 2D

# Graded Symmetry Groups

AGACSE2021

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

any $$k$$, even and odd

any # dimensions,

any metric

closed form exp and log. $$e^{a+b} = e^ae^b$$ iff $$ab = ba$$

2. Geometry | coordinate, dimension and metric agnostic

this is a vector.

$$x^2 = \lVert x \rVert^2$$

When a second non colinear vector is added

• we can measure two lengths
• we can measure one angle
• we know our space is >= 2D

We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'

# Graded Symmetry Groups

AGACSE2021

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

any $$k$$, even and odd

any # dimensions,

any metric

closed form exp and log. $$e^{a+b} = e^ae^b$$ iff $$ab = ba$$

2. Geometry | coordinate, dimension and metric agnostic

this is a vector.

$$x^2 = \lVert x \rVert^2$$

When a second non colinear vector is added

• we can measure two lengths
• we can measure one angle
• we know our space is >= 2D

We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'

Independent of the choice of basis, our vectors, bivectors and n-vectors are still attached to the origin. To break free from the origin in our space of interest we projectivize and move it outside.

# Graded Symmetry Groups

AGACSE2021

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

any $$k$$, even and odd

any # dimensions,

any metric

closed form exp and log. $$e^{a+b} = e^ae^b$$ iff $$ab = ba$$

2. Geometry | coordinate, dimension and metric agnostic

this is a vector.

• we know our space is >= 2D

We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'

Independent of the choice of basis, our vectors, bivectors and n-vectors are still attached to the origin. To break free from the origin in our space of interest we projectivize and move it outside.

This works for spaces of arbitrary (constant) curvature.

# Graded Symmetry Groups

AGACSE2021

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

any $$k$$, even and odd

any # dimensions,

any metric

closed form exp and log. $$e^{a+b} = e^ae^b$$ iff $$ab = ba$$

2. Geometry | coordinate, dimension and metric agnostic

this is a vector.

• we know our space is >= 2D

We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'

Independent of the choice of basis, our vectors, bivectors and n-vectors are still attached to the origin. To break free from the origin in our space of interest we projectivize and move it outside.

This works for spaces of arbitrary (constant) curvature.

# Graded Symmetry Groups

AGACSE2021

a 4D homogeneous point

a 6D Plucker line

a 4D homogeneous plane

any $$k$$, even and odd

any # dimensions,

any metric

closed form exp and log. $$e^{a+b} = e^ae^b$$ iff $$ab = ba$$

2. Geometry | coordinate, dimension and metric agnostic

this is a vector.

• we know our space is >= 2D

We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'

Independent of the choice of basis, our vectors, bivectors and n-vectors are still attached to the origin. To break free from the origin in our space of interest we projectivize and move it outside.

This works for spaces of arbitrary (constant) curvature.

# Graded Symmetry Groups

AGACSE2021

2. Geometry | coordinate, dimension and metric agnostic

• we know our space is >= 2D

We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'

Independent of the choice of basis, our vectors, bivectors and n-vectors are still attached to the origin. To break free from the origin in our space of interest we projectivize and move it outside.

This works for spaces of arbitrary (constant) curvature.

Similarly, geometric elements can be defined in a coordinate, metric and dimension agnostic way, and only impose a lower limit on the dimensionality of the space.

# Graded Symmetry Groups

AGACSE2021

2. Geometry | coordinate, dimension and metric agnostic

• we know our space is >= 2D

We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'

Similarly, geometric elements can be defined in a coordinate, metric and dimension agnostic way, and only impose a lower limit on the dimensionality of the space.

However, formulating a problem independent of the dimensionality of the space does require some extra care...

# Graded Symmetry Groups

AGACSE2021

2. Geometry | coordinate, dimension and metric agnostic

• we know our space is >= 2D

We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'

Similarly, geometric elements can be defined in a coordinate, metric and dimension agnostic way, and only impose a lower limit on the dimensionality of the space.

However, formulating a problem independent of the dimensionality of the space does require some extra care...

robot arm

floor plan

robot arm

Both 2D drawings have points and lines. What happens if we go from two to three dimensions?

# Graded Symmetry Groups

AGACSE2021

2. Geometry | coordinate, dimension and metric agnostic

• we know our space is >= 2D

We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'

Similarly, geometric elements can be defined in a coordinate, metric and dimension agnostic way, and only impose a lower limit on the dimensionality of the space.

However, formulating a problem independent of the dimensionality of the space does require some extra care...

robot arm

floor plan

robot arm

Both 2D drawings have points and lines. What happens if we go from two to three dimensions?

When going to 3D, the lines on the left stay lines. Those on the right become planes .. working dimension independent means embracing duality!

# Graded Symmetry Groups

AGACSE2021

2. Geometry | coordinate, dimension and metric agnostic

• we know our space is >= 2D

We can now repeat this ad nauseam. This is of course exactly the structure of geometric algebra. It allows us to postpone the choice of basis till the latest possible time. Such an approach is commonly called 'coordinate free'

Similarly, geometric elements can be defined in a coordinate, metric and dimension agnostic way, and only impose a lower limit on the dimensionality of the space.

However, formulating a problem independent of the dimensionality of the space does require some extra care...

robot arm

floor plan

robot arm

When going to 3D, the lines on the left stay lines. Those on the right become planes .. working dimension independent means embracing duality!

points and lines defined dually

points and lines defined directly

# Graded Symmetry Groups

AGACSE2021

2. Geometry | Forques !

However, formulating a problem independent of the dimensionality of the space does require some extra care...

robot arm

floor plan

robot arm

When going to 3D, the lines on the left stay lines. Those on the right become planes .. working dimension independent means embracing duality!

points and lines defined dually

points and lines defined directly

# Graded Symmetry Groups

AGACSE2021

2. Geometry | Forques !

In the body frame :

push at origin = translation

push at infinity = rotation

Force and Torque are both lines.

in any # dimensions.

# Graded Symmetry Groups

AGACSE2021

2. Geometry | Forques !

In the body frame :

push at origin = translation

push at infinity = rotation

Force and Torque are both lines.

in any # dimensions.

nobody in their right mind does live programming during a talk.

Unfortunately for you all, that leaves me qualified.

#### Graded Symmetry Groups

By Steven De Keninck

• 572