Duals - Revision history

Eric Lengyel at 19:57, 25 January 2026

2026-01-25T19:57:00Z

← Older revision		Revision as of 19:57, 25 January 2026
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	The right bulk dual produces results equivalent to the Hodge star operator and thus satisfies the identity		The right bulk dual produces results equivalent to the Hodge star operator and thus satisfies the identity

	:$$\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605} = (\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b){\large\unicode{x1D7D9}}$$.		:$$\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605} = (\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b){\large\unicode{x1D7D9}}$$

			when $$\mathbf a$$ and $$\mathbf b$$ have the same grade.

	== Antidual ==		== Antidual ==

Eric Lengyel at 08:49, 22 December 2024

2024-12-22T08:49:34Z

← Older revision		Revision as of 08:49, 22 December 2024
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	== Dual ==		== Dual ==

	The ''metric dual'' or just "dual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2605}$$ and defined as		The (right) ''metric dual'' or just "dual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2605}$$ and defined as

	:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf{Gu}}$$ ,		:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf{Gu}}$$ ,
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	:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \tilde{\mathbf u} \mathbin{\unicode{x27D1}} {\large\unicode{x1D7D9}}$$ .		:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \tilde{\mathbf u} \mathbin{\unicode{x27D1}} {\large\unicode{x1D7D9}}$$ .

			The right bulk dual produces results equivalent to the Hodge star operator and thus satisfies the identity

			:$$\mathbf a \wedge \mathbf b^\unicode["segoe ui symbol"]{x2605} = (\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b){\large\unicode{x1D7D9}}$$.

	== Antidual ==		== Antidual ==

	The ''metric antidual'' or just "antidual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2606}$$ and defined as		The (right) ''metric antidual'' or just "antidual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2606}$$ and defined as

	:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,		:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,
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	== Duals of Basis Elements ==		== Duals of Basis Elements ==

	The following table lists the bulk and weight duals for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$.		The following table lists the bulk and weight duals for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$. In addition to the right duals, there are also left bulk and weight duals denoted by $$\mathbf u_\unicode["segoe ui symbol"]{x2605}$$ and $$\mathbf u_\unicode["segoe ui symbol"]{x2606}$$ for which the right complement operation is replaced by the left complement operation.

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

			Taking either dual twice causes the sign to change according to the formula

			:$$\mathbf u^{\unicode["segoe ui symbol"]{x2605}\unicode["segoe ui symbol"]{x2605}} = \mathbf u^{\unicode["segoe ui symbol"]{x2606}\unicode["segoe ui symbol"]{x2606}} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}\,\det(\mathfrak g)\mathbf u$$ .

			Since the determinant of the [[metric]] $$\mathfrak g$$ is zero in the projective algebra $$\mathcal G_{3,0,1}$$, applying either dual twice always produces zero here.

	== Duals of Geometries ==		== Duals of Geometries ==

Eric Lengyel at 01:21, 8 July 2024

2024-07-08T01:21:28Z

← Older revision		Revision as of 01:21, 8 July 2024
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	where $$\mathbf G$$ is the $$16 \times 16$$ [[metric exomorphism matrix]]. In projective geometric algebra, this dual is also called the ''bulk dual'' because it is the [[complement]] of the bulk components, as expressed by		where $$\mathbf G$$ is the $$16 \times 16$$ [[metric exomorphism matrix]]. In projective geometric algebra, this dual is also called the ''bulk dual'' because it is the [[complement]] of the bulk components, as expressed by

	:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf u_\unicode{x25CF}}$$ .		:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf u_\unicode["segoe ui symbol"]{x25CF}}$$ .

	The bulk dual satisfies the following identity based on the [[geometric product]]:		The bulk dual satisfies the following identity based on the [[geometric product]]:
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	where $$\mathbb G$$ is the $$16 \times 16$$ [[metric antiexomorphism matrix]]. In projective geometric algebra, this dual is also called the ''weight dual'' because it is the [[complement]] of the weight components, as expressed by		where $$\mathbb G$$ is the $$16 \times 16$$ [[metric antiexomorphism matrix]]. In projective geometric algebra, this dual is also called the ''weight dual'' because it is the [[complement]] of the weight components, as expressed by

	:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbf u_\unicode{x25CB}}$$ .		:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbf u_\unicode["segoe ui symbol"]{x25CB}}$$ .

	The weight dual satisfies the following identity based on the [[geometric antiproduct]]:		The weight dual satisfies the following identity based on the [[geometric antiproduct]]:

Eric Lengyel at 01:33, 12 May 2024

2024-05-12T01:33:21Z

← Older revision		Revision as of 01:33, 12 May 2024
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	:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} \mathbin{\unicode{x27C7}} \mathbf 1$$ .		:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} \mathbin{\unicode{x27C7}} \mathbf 1$$ .

			== Duals of Basis Elements ==

	The following table lists the bulk and weight duals for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$.		The following table lists the bulk and weight duals for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$.
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	[[Image:Duals.svg\|720px]]		[[Image:Duals.svg\|720px]]

	== Geometries ==		== Duals of Geometries ==

	The bulk duals and weight duals of geometries in the 4D rigid geometric algebra are listed in the following table.		The bulk duals and weight duals of geometries in the 4D rigid geometric algebra are listed in the following table.

Eric Lengyel at 01:31, 12 May 2024

2024-05-12T01:31:27Z

← Older revision		Revision as of 01:31, 12 May 2024
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	:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} \mathbin{\unicode{x27C7}} \mathbf 1$$ .		:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} \mathbin{\unicode{x27C7}} \mathbf 1$$ .

			The following table lists the bulk and weight duals for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$.

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

	== Geometries ==		== Geometries ==

Eric Lengyel at 23:32, 13 April 2024

2024-04-13T23:32:14Z

← Older revision		Revision as of 23:32, 13 April 2024
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	\| style="padding: 12px;" \| $$\mathbf g^\unicode["segoe ui symbol"]{x2606} = -g_x \mathbf e_1 - g_y \mathbf e_2 - g_z \mathbf e_3$$		\| style="padding: 12px;" \| $$\mathbf g^\unicode["segoe ui symbol"]{x2606} = -g_x \mathbf e_1 - g_y \mathbf e_2 - g_z \mathbf e_3$$
	\|}		\|}

			== In the Book ==

			* Duals are introduced in Section 2.12.

	== See Also ==		== See Also ==
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	* [[Complements]]		* [[Complements]]
	* [[Bulk and weight]]		* [[Bulk and weight]]

	~~== In the Book ==~~

	* Duals are introduced in Section 2.12.

Eric Lengyel at 05:05, 13 April 2024

2024-04-13T05:05:57Z

@@ Line 28: / Line 28: @@
 :$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} \mathbin{\unicode{x27C7}} \mathbf 1$$ .
 == See Also ==
@@ Line 33: / Line 53: @@
 * [[Complements]]
 * [[Bulk and weight]]

Eric Lengyel at 01:54, 13 April 2024

2024-04-13T01:54:47Z

← Older revision		Revision as of 01:54, 13 April 2024
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	:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf{Gu}}$$ ,		:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf{Gu}}$$ ,

	where $$\mathbf G$$ is the $$16 \times 16$$ metric exomorphism matrix. In projective geometric algebra, this dual is also called the ''bulk dual'' because it is the [[complement]] of the bulk components, as expressed by		where $$\mathbf G$$ is the $$16 \times 16$$ [[metric exomorphism matrix]]. In projective geometric algebra, this dual is also called the ''bulk dual'' because it is the [[complement]] of the bulk components, as expressed by

	:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf u_\unicode{x25CF}}$$ .		:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf u_\unicode{x25CF}}$$ .
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	:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,		:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,

	where $$\mathbb G$$ is the $$16 \times 16$$ metric antiexomorphism matrix. In projective geometric algebra, this dual is also called the ''weight dual'' because it is the [[complement]] of the weight components, as expressed by		where $$\mathbb G$$ is the $$16 \times 16$$ [[metric antiexomorphism matrix]]. In projective geometric algebra, this dual is also called the ''weight dual'' because it is the [[complement]] of the weight components, as expressed by

	:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbf u_\unicode{x25CB}}$$ .		:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbf u_\unicode{x25CB}}$$ .

Eric Lengyel at 01:36, 13 April 2024

2024-04-13T01:36:54Z

← Older revision		Revision as of 01:36, 13 April 2024
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	:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf{Gu}}$$ ,		:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf{Gu}}$$ ,

	where $$\mathbf G$$ is the ~~extended~~ metric ~~tensor~~. In projective geometric algebra, this dual is also called the ''bulk dual'' because it is the [[complement]] of the bulk components, as expressed by		where $$\mathbf G$$ is the $$16 \times 16$$ metric exomorphism matrix. In projective geometric algebra, this dual is also called the ''bulk dual'' because it is the [[complement]] of the bulk components, as expressed by

	:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf u_\unicode{x25CF}}$$ .		:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf u_\unicode{x25CF}}$$ .
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	:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,		:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,

	where $$\mathbb G$$ is the ~~extended antimetric tensor~~. In projective geometric algebra, this dual is also called the ''weight dual'' because it is the [[complement]] of the weight components, as expressed by		where $$\mathbb G$$ is the $$16 \times 16$$ metric antiexomorphism matrix. In projective geometric algebra, this dual is also called the ''weight dual'' because it is the [[complement]] of the weight components, as expressed by

	:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbf u_\unicode{x25CB}}$$ .		:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbf u_\unicode{x25CB}}$$ .

Eric Lengyel: /* See Also */

2024-04-12T06:26:29Z

← Older revision		Revision as of 06:26, 12 April 2024
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	* [[Complements]]		* [[Complements]]
	* [[Bulk and ~~Weight~~]]		* [[Bulk and weight]]