Geometric products - Revision history

Eric Lengyel at 22:21, 6 February 2026

2026-02-06T22:21:52Z

← Older revision		Revision as of 22:21, 6 February 2026
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	The geometric product is equal to the sum of the [[transwedge products]] of orders 0 through $$n$$, where $$n$$ is the dimension of the algebra, in this case 4.		The geometric product is equal to the sum of the [[transwedge products]] of orders 0 through $$n$$, where $$n$$ is the dimension of the algebra, in this case 4.

			When $$\mathbf a$$ is a vector, the geometric product $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ can be decomposed as

			:$$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf b \vee \mathbf a^{\unicode["segoe ui symbol"]{x2605}}$$,

			and the geometric product $$\mathbf b \mathbin{\unicode{x27D1}} \mathbf a$$ can be decomposed as

			:$$\mathbf b \mathbin{\unicode{x27D1}} \mathbf a = \mathbf b \wedge \mathbf a + \mathbf a_{\unicode["segoe ui symbol"]{x2605}} \vee \mathbf b$$.

			If $$\mathbf a$$ and $$\mathbf b$$ are both vectors, then both of these decompositions reduce to

			:$$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \cdot \mathbf b$$.

	The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.		The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.

Eric Lengyel at 20:28, 20 May 2025

2025-05-20T20:28:52Z

← Older revision		Revision as of 20:28, 20 May 2025
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	The ''geometric product'' is ~~the fundamental product~~ of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct.		The ''geometric product'' is a method of multiplication in geometric algebra derived from the [[exterior product]] and the [[metric]]. There are two products with symmetric properties called the geometric product and geometric antiproduct.

	== Geometric Product ==		== Geometric Product ==
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	:$$\mathbf e_i \mathbin{\unicode{x27D1}} \mathbf e_j = \mathbf e_i \wedge \mathbf e_j$$, for $$i \neq j$$.		:$$\mathbf e_i \mathbin{\unicode{x27D1}} \mathbf e_j = \mathbf e_i \wedge \mathbf e_j$$, for $$i \neq j$$.

			The geometric product is equal to the sum of the [[transwedge products]] of orders 0 through $$n$$, where $$n$$ is the dimension of the algebra, in this case 4.

	The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.		The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.
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	== Geometric Antiproduct ==		== Geometric Antiproduct ==

	The geometric antiproduct is a dual to the geometric product. The geometric antiproduct between elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x27C7}} \mathbf b$$ and is read as "$$\mathbf a$$ antiwedge-dot $$\mathbf b$$".		The geometric antiproduct is dual to the geometric product. The geometric antiproduct between elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x27C7}} \mathbf b$$ and is read as "$$\mathbf a$$ antiwedge-dot $$\mathbf b$$".

	The same metric that defines products of basis vectors under the geometric product also applies to the geometric antiproduct, except that now it defines products of basis [[antivectors]]. Three basis antivectors square to $$+{\large\unicode{x1D7D9}}$$, zero basis antivectors square to $$-{\large\unicode{x1D7D9}}$$, and one basis antivector squares to 0. The geometric antiproduct between two different basis antivectors is given by the [[antiwedge product]]. We can write these rules as follows.		The same metric that defines products of basis vectors under the geometric product also applies to the geometric antiproduct, except that now it defines products of basis [[antivectors]]. Three basis antivectors square to $$+{\large\unicode{x1D7D9}}$$, zero basis antivectors square to $$-{\large\unicode{x1D7D9}}$$, and one basis antivector squares to 0. The geometric antiproduct between two different basis antivectors is given by the [[antiwedge product]]. We can write these rules as follows.

Eric Lengyel: /* See Also */

2025-05-20T19:56:06Z

Eric Lengyel at 23:25, 13 April 2024

2024-04-13T23:25:20Z

← Older revision		Revision as of 23:25, 13 April 2024
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	== De Morgan Laws ==		== De Morgan Laws ==

	~~There are many possible geometric products and antiproducts. The signs of the results they produce differ in grade-dependent ways, but are otherwise equivalent.~~ The relationship between the product and antiproduct is ~~fixed by a specific choice~~ of ~~dualization function that exchanges~~ full and empty dimensions~~. We choose the left and right [[complements]] as the dualization function and its inverse~~. We can then express each product in terms of the other through an analog of De Morgan's laws as follows.		The relationship between the product and antiproduct is based on an exchange of full and empty dimensions. We can then express each product in terms of the other through an analog of De Morgan's laws as follows.

	:$$\overline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \overline{\mathbf b}$$		:$$\overline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \overline{\mathbf b}$$
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	:$$\underline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \underline{\mathbf b}$$		:$$\underline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \underline{\mathbf b}$$

			== In the Book ==

			* The geometric product and antiproduct are introduced in Section 3.1.

	== See Also ==		== See Also ==

Eric Lengyel at 02:08, 23 January 2024

2024-01-23T02:08:21Z

← Older revision		Revision as of 02:08, 23 January 2024
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	Cells highlighted in green correspond to the contribution from the [[wedge product]].		Cells highlighted in green correspond to the contribution from the [[wedge product]].

	~~<span style="font-size: 150%;">Geometric Product $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$</span>~~

	[[Image:GeometricProduct.svg\|720px]]		[[Image:GeometricProduct.svg\|720px]]
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	Cells highlighted in green correspond to the contribution from the [[antiwedge product]].		Cells highlighted in green correspond to the contribution from the [[antiwedge product]].

	~~<span style="font-size: 150%;">Geometric Antiproduct $$\mathbf a \mathbin{\unicode{x27C7}} \mathbf b$$</span>~~

	[[Image:GeometricAntiproduct.svg\|720px]]		[[Image:GeometricAntiproduct.svg\|720px]]

Eric Lengyel at 21:55, 21 January 2024

2024-01-21T21:55:08Z

← Older revision		Revision as of 21:55, 21 January 2024
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	The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.		The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.

	Cells ~~colored yellow~~ correspond to the contribution from the [[wedge product]].		Cells highlighted in green correspond to the contribution from the [[wedge product]].

	<span style="font-size: 150%;">Geometric Product $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$</span>		<span style="font-size: 150%;">Geometric Product $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$</span>
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	The following Cayley table shows the geometric antiproducts between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric antiproduct is the [[antiscalar]] basis element $${\large\unicode{x1D7D9}}$$.		The following Cayley table shows the geometric antiproducts between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric antiproduct is the [[antiscalar]] basis element $${\large\unicode{x1D7D9}}$$.

	Cells ~~colored yellow~~ correspond to the contribution from the [[antiwedge product]].		Cells highlighted in green correspond to the contribution from the [[antiwedge product]].

	<span style="font-size: 150%;">Geometric Antiproduct $$\mathbf a \mathbin{\unicode{x27C7}} \mathbf b$$</span>		<span style="font-size: 150%;">Geometric Antiproduct $$\mathbf a \mathbin{\unicode{x27C7}} \mathbf b$$</span>

Eric Lengyel at 21:54, 21 January 2024

2024-01-21T21:54:31Z

← Older revision		Revision as of 21:54, 21 January 2024
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	Cells colored yellow correspond to the contribution from the [[wedge product]].		Cells colored yellow correspond to the contribution from the [[wedge product]].

			<span style="font-size: 150%;">Geometric Product $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$</span>

	[[Image:GeometricProduct.svg\|720px]]		[[Image:GeometricProduct.svg\|720px]]
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	Cells colored yellow correspond to the contribution from the [[antiwedge product]].		Cells colored yellow correspond to the contribution from the [[antiwedge product]].

			<span style="font-size: 150%;">Geometric Antiproduct $$\mathbf a \mathbin{\unicode{x27C7}} \mathbf b$$</span>

	[[Image:GeometricAntiproduct.svg\|720px]]		[[Image:GeometricAntiproduct.svg\|720px]]

Eric Lengyel at 00:41, 26 August 2023

2023-08-26T00:41:01Z

← Older revision		Revision as of 00:41, 26 August 2023
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	The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.		The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.

	Cells colored yellow correspond to the contribution from the [[wedge ~~product]], and cells colored orange correspond to the contribution from the [[dot~~ product]].		Cells colored yellow correspond to the contribution from the [[wedge product]].


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	The following Cayley table shows the geometric antiproducts between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric antiproduct is the [[antiscalar]] basis element $${\large\unicode{x1D7D9}}$$.		The following Cayley table shows the geometric antiproducts between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric antiproduct is the [[antiscalar]] basis element $${\large\unicode{x1D7D9}}$$.

	Cells colored yellow correspond to the contribution from the [[antiwedge ~~product]], and cells colored orange correspond to the contribution from the [[antidot~~ product]].		Cells colored yellow correspond to the contribution from the [[antiwedge product]].

Eric Lengyel: Created page with "The ''geometric product'' is the fundamental product of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct. == Geometric Product == The geometric product between two elements $$\mathbf a$$ and $$\mathbf b$$ has often been written by simply juxtaposing its operands as $$\mathbf{ab}$$ without the use of any infix operator. However, this clearly becomes impractical when both a product and antiproduct..."

2023-07-15T06:29:36Z

Created page with "The ''geometric product'' is the fundamental product of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct. == Geometric Product == The geometric product between two elements $$\mathbf a$$ and $$\mathbf b$$ has often been written by simply juxtaposing its operands as $$\mathbf{ab}$$ without the use of any infix operator. However, this clearly becomes impractical when both a product and antiproduct..."

New page

The ''geometric product'' is the fundamental product of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct.

== Geometric Product ==

The geometric product between two elements $$\mathbf a$$ and $$\mathbf b$$ has often been written by simply juxtaposing its operands as $$\mathbf{ab}$$ without the use of any infix operator. However, this clearly becomes impractical when both a product and antiproduct are present in the same context, which is now known to be necessary for a proper understanding of the algebra. To remedy the situation, we write the geometric product between elements $$\mathbf a$$ and $$\mathbf b$$ as $$\mathbf a \mathbin{\unicode{x27D1}} \mathbf b$$ and read it as "$$\mathbf a$$ wedge-dot $$\mathbf b$$".

The geometric product is characterized by a metric that defines the products of the basis vectors with themselves. The subscript in $$\mathcal G_{3,0,1}$$ means that three basis vectors square to +'''1''', zero basis vectors square to −'''1''', and one basis vector squares to 0. The geometric product between two different basis vectors is given by the [[wedge product]]. We can write these rules as follows.

:$$\mathbf e_1 \mathbin{\unicode{x27D1}} \mathbf e_1 = \mathbf 1$$

:$$\mathbf e_2 \mathbin{\unicode{x27D1}} \mathbf e_2 = \mathbf 1$$

:$$\mathbf e_3 \mathbin{\unicode{x27D1}} \mathbf e_3 = \mathbf 1$$

:$$\mathbf e_4 \mathbin{\unicode{x27D1}} \mathbf e_4 = 0$$

:$$\mathbf e_i \mathbin{\unicode{x27D1}} \mathbf e_j = \mathbf e_i \wedge \mathbf e_j$$, for $$i \neq j$$.

The following Cayley table shows the geometric products between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.

Cells colored yellow correspond to the contribution from the [[wedge product]], and cells colored orange correspond to the contribution from the [[dot product]].

[[Image:GeometricProduct.svg|720px]]

== Geometric Antiproduct ==

The geometric antiproduct is a dual to the geometric product. The geometric antiproduct between elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x27C7}} \mathbf b$$ and is read as "$$\mathbf a$$ antiwedge-dot $$\mathbf b$$".

The same metric that defines products of basis vectors under the geometric product also applies to the geometric antiproduct, except that now it defines products of basis [[antivectors]]. Three basis antivectors square to $$+{\large\unicode{x1D7D9}}$$, zero basis antivectors square to $$-{\large\unicode{x1D7D9}}$$, and one basis antivector squares to 0. The geometric antiproduct between two different basis antivectors is given by the [[antiwedge product]]. We can write these rules as follows.

:$$\overline{\mathbf e_1} \mathbin{\unicode{x27C7}} \overline{\mathbf e_1} = {\large\unicode{x1D7D9}}$$

:$$\overline{\mathbf e_2} \mathbin{\unicode{x27C7}} \overline{\mathbf e_2} = {\large\unicode{x1D7D9}}$$

:$$\overline{\mathbf e_3} \mathbin{\unicode{x27C7}} \overline{\mathbf e_3} = {\large\unicode{x1D7D9}}$$

:$$\overline{\mathbf e_4} \mathbin{\unicode{x27C7}} \overline{\mathbf e_4} = 0$$

:$$\overline{\mathbf e_i} \mathbin{\unicode{x27C7}} \overline{\mathbf e_j} = \overline{\mathbf e_i} \vee \overline{\mathbf e_j}$$, for $$i \neq j$$.

The following Cayley table shows the geometric antiproducts between all pairs of basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. The identity of the geometric antiproduct is the [[antiscalar]] basis element $${\large\unicode{x1D7D9}}$$.

Cells colored yellow correspond to the contribution from the [[antiwedge product]], and cells colored orange correspond to the contribution from the [[antidot product]].

[[Image:GeometricAntiproduct.svg|720px]]

== De Morgan Laws ==

There are many possible geometric products and antiproducts. The signs of the results they produce differ in grade-dependent ways, but are otherwise equivalent. The relationship between the product and antiproduct is fixed by a specific choice of dualization function that exchanges full and empty dimensions. We choose the left and right [[complements]] as the dualization function and its inverse. We can then express each product in terms of the other through an analog of De Morgan's laws as follows.

:$$\overline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \overline{\mathbf b}$$

:$$\overline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \overline{\mathbf b}$$

:$$\underline{\mathbf a \mathbin{\smash{\unicode{x27D1}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27C7}} \underline{\mathbf b}$$

:$$\underline{\mathbf a \mathbin{\smash{\unicode{x27C7}}} \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \mathbin{\unicode{x27D1}} \underline{\mathbf b}$$

== See Also ==

* [[Wedge products]]
* [[Dot products]]
* [[Complements]]

← Older revision		Revision as of 19:56, 20 May 2025
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	== See Also ==		== See Also ==

	* [[~~Wedge~~ products]]		* [[Transwedge products]]
			* [[Exterior products]]
	* [[Dot products]]		* [[Dot products]]
	* [[Complements]]		* [[Complements]]