Blades in Geometric Algebra

This page is a sub-page of our page on Geometric Algebra.

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GA videos: http://portail-video.univ-lr.fr/AGACSE-2012
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The sub-pages of this page are:

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Related KMR-pages:

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Other relevant sources of information:

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Formulas from Ambjörn’s presentation at SIGGRAPH 2000:

\, \mathrm{V}^n \, , \, e_1 \, , e_2 \, , \dots , e_n \,

\, \mathrm{G} = \mathrm{G}_n = \mathrm{G}(\mathrm{V}^n) \,

\, M = \sum\limits_{k = 0}^{n} {\langle \, M \, \rangle}_k \,

\, {\langle \, M \, \rangle}_k = B_1 + B_2 + \cdots \,

\, A_k = a_1 \wedge a_2 \wedge \cdots \wedge a_k \,

\, A_k \neq 0 \, \iff \, \{ a_1, a_2, \dots , a_k \} \,

\, P \in \mathrm{G}_n \,


\, P = p_1 \wedge p_2 \wedge \cdots \wedge p_n \,


\, I = e_1 \wedge e_2 \wedge \cdots \wedge e_n \,

\, [P] = PI^{-1} \,


\, \text{dual}(X) = XI^{-1} \,

\, \text{dual}(X) = X^* \,

\, B = b_1 \wedge b_2 \wedge \cdots \wedge b_m \,

\, \overline{B} \subseteq \mathrm{V}^n \,

\, \overline{B} = \text{Linspan}\{ b_1, b_2, \dots , b_m \} = \,

\, = \text{Linspan}\{ b \in \mathrm{G}_n : b \wedge B = 0 \} \,

\, \{ e_1, \dots , e_m \} \,

\, b_i = \sum\limits_{k = 0}^{m} b_{ik}e_k \; \text{for} \; i = 1, \, \dots \, , m \; , \,

\, B =( \det{b_{ik}}) \, e_1 \wedge e_2 \wedge \cdots \wedge e_m = \,

\, = (\det{b_{ik}}) \, e_1 e_2 \cdots e_m = \,

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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The Story of Geometric Algebra

Every kid knows that the result of applying Socks to Shoes is different from the result of applying Shoes to Socks. In mathematics we express this difference as Socks (Shoes) ≠ Shoes (Socks), which is read as “Socks operating on Shoes is not equal to Shoes operating on Socks”, and we say that the operations of putting on Socks and Shoes are sensitive to the order in which they are applied. We refer to such operations as non-commutative.

In contrast, the first operations the we encounter when learning mathematics are insensitive to the order in which they are applied. They are commutative as we say. For example, 3+5 = 5+3 and 4 x 6 = 6 x 4. In fact, the first non-commutative operation that we encounter in math education is multiplication of matrices, and this encounter normally takes place in a course called linear algebra during the first year of university studies. No wonder then that the concept of non-commutativity is considered difficult to understand by many students.

And yet, within geometry, the result of applying two operations are in general sensitive to the order in which these operations are applied. If, for example, we slide a figure a certain distance along a certain direction, and then rotate the resulting figure around a certain point, then the resulting position of the figure in general differs from the position that appears if we apply the sliding operation and the rotation operation of the same figure in the reverse order.

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“You can’t add apples and pears” is a familiar statement of school mathematics. But, in fact, you can. When you’re adding apples and pears, just as you do when you are bringing a bag of apples and pears home from the grocery store, you are just using a different form of addition (called formal addition or direct addition in mathematics).

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Types of Blades:

0-blades
1-blades
2-blades
3-blades

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0-blades

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1-blades

Equivalent representations of a 1-blade in 2D:

The interactive simulation that created this movie.

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2-blades

Adding two 2-blades in 3D:

The interactive simulation that created this movie.

Equivalent representations of a 2-blade in 2D (wedge and triangular):

The interactive simulation that created this movie.

Equivalent elliptic shapes of a 2-blade in 2D:

The interactive simulation that created this movie.

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3-blades

Equivalent representations of a 3-blade in 3D:

The interactive simulation that created this movie.

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