Transwedge products - Revision history

Eric Lengyel at 09:12, 17 December 2025

2025-12-17T09:12:56Z

← Older revision		Revision as of 09:12, 17 December 2025
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	For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it’s nonzero.		For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it’s nonzero.

	The sum of all possible transwedge products, negating for orders 2 and 3 modulo 4, yields the geometric product. That is,		The sum of all possible transwedge products, negating for orders 2 and 3 modulo 4, yields the [[geometric product]]. That is,

	:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,		:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,
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	where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], and we are now summing over all basis elements of [[antigrade]] $$k$$.		where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], and we are now summing over all basis elements of [[antigrade]] $$k$$.

	The sum of all possible transwedge antiproducts, again negating for orders 2 and 3 modulo 4, yields the geometric antiproduct. That is,		The sum of all possible transwedge antiproducts, again negating for orders 2 and 3 modulo 4, yields the [[geometric antiproduct]]. That is,

	:$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,		:$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,

Eric Lengyel at 20:54, 26 May 2025

2025-05-26T20:54:30Z

← Older revision		Revision as of 20:54, 26 May 2025
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	For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have		For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have

	:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b - \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,		:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b - \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf{\tilde b}$$,

	where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf~~{\tilde~~ a}^{\unicode{x2605}}$$, but this time with a reversal to account for the sign change. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.		where the [[dot product]] is again attributable to the right contraction $$\mathbf{\tilde b} \vee \mathbf a^{\unicode{x2605}}$$, but this time with a reversal to account for the sign change. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.

	== Transwedge Antiproduct ==		== Transwedge Antiproduct ==

Eric Lengyel at 20:51, 26 May 2025

2025-05-26T20:51:57Z

← Older revision		Revision as of 20:51, 26 May 2025
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	where $$n$$ is the dimension of the algebra.		where $$n$$ is the dimension of the algebra.

	~~The~~ [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)		When the [[metric]] is diagonal, the [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)

Eric Lengyel at 22:02, 21 May 2025

2025-05-21T22:02:46Z

← Older revision		Revision as of 22:02, 21 May 2025
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	:$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.		:$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.

	When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, which is an interior product. If $$\mathbf a$$ and $$\mathbf b$$ have the same grade, then this interior product reduces to the inner product $$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$. If $$k > \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.		When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, which is an interior product. If $$\mathbf a$$ and $$\mathbf b$$ have the same grade, then this interior product reduces to the inner product $$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$.

			If $$k > \operatorname{gr}(\mathbf a)$$ or $$k > \operatorname{gr}(\mathbf b)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.

	An equivalent definition for the transwedge product is given by		An equivalent definition for the transwedge product is given by

Eric Lengyel at 22:01, 21 May 2025

2025-05-21T22:01:21Z

← Older revision		Revision as of 22:01, 21 May 2025
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	:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b - \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,		:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b - \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,

	where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.		where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$, but this time with a reversal to account for the sign change. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.

	== Transwedge Antiproduct ==		== Transwedge Antiproduct ==

Eric Lengyel at 21:26, 21 May 2025

2025-05-21T21:26:10Z

← Older revision		Revision as of 21:26, 21 May 2025
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	:$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.		:$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.

	When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, which is an interior product. If $$k > \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.		When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, which is an interior product. If $$\mathbf a$$ and $$\mathbf b$$ have the same grade, then this interior product reduces to the inner product $$\mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$. If $$k > \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.

	An equivalent definition for the transwedge product is given by		An equivalent definition for the transwedge product is given by

Eric Lengyel at 21:10, 21 May 2025

2025-05-21T21:10:05Z

Eric Lengyel: /* Transwedge Antiproduct */

2025-05-20T00:50:16Z

Transwedge Antiproduct

← Older revision		Revision as of 00:50, 20 May 2025
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	:$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,		:$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,

	The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, and 3.		The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, 3, and 4.

Eric Lengyel at 00:49, 20 May 2025

2025-05-20T00:49:53Z

← Older revision		Revision as of 00:49, 20 May 2025
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	where $$n$$ is the dimension of the algebra.		where $$n$$ is the dimension of the algebra.

	The [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, and 3. (~~The transwedge product~~ of ~~order 4 is~~ zero in this algebra due to the degenerate metric.)		The [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, 3, and 4. (Some of the products are zero in this algebra due to the degenerate metric.)

Eric Lengyel at 00:06, 20 May 2025

2025-05-20T00:06:19Z

← Older revision		Revision as of 00:06, 20 May 2025
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	where $$n$$ is the dimension of the algebra.		where $$n$$ is the dimension of the algebra.

	The [[geometric product]] of each pair of the 16 basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ ~~is given by exactly one of the transwedge products. These are shown~~ in the following table, which color codes the transwedge products of order 0, 1, 2, and 3. (The transwedge product of order 4 is zero in this algebra due to the degenerate metric.)		The [[geometric product]] of each pair of the basis elements in a geometric algebra is given by exactly one of the transwedge products. These are shown for the 16 basis elements of the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ in the following table, which color codes the transwedge products of order 0, 1, 2, and 3. (The transwedge product of order 4 is zero in this algebra due to the degenerate metric.)

← Older revision		Revision as of 21:10, 21 May 2025
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	The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as		The transwedge product of order $$k$$ is written with a double upward wedge having either a subscript or an underscript indicating the order. It is defined as

	:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf~~{\tilde~~ c^{\unicode{x2605}}})}$$.		:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf{\underline c} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf c^{\unicode{x2605}})}$$.

	The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,		The set $$\mathcal B_k$$ is the set of all basis elements having grade $$k$$. When $$k = 0$$, this set is simply $$\{\mathbf 1\}$$, and the transwedge product reduces to the wedge (exterior) product. That is,
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	:$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.		:$$\displaystyle\mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b$$.

	When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf~~{\tilde~~ a^{\unicode{x2605}}}$$, which is an interior product. If $$k > \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.		When $$k = \operatorname{gr}(\mathbf a)$$, the transwedge product reduces to the right [[contraction]] $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$, which is an interior product. If $$k > \operatorname{gr}(\mathbf a)$$, then $$\mathbf a \mathbin{\unicode{x2A53}_k} \mathbf b = 0$$.

	An equivalent definition for the transwedge product is given by		An equivalent definition for the transwedge product is given by

	:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf~~{\tilde~~ c_{\unicode{x2605}}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,		:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_k}{(\mathbf c_{\unicode{x2605}} \vee \mathbf a) \wedge (\mathbf b \vee \mathbf{\overline c})}$$,

	where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.		where the right contraction on the right side of the summand is now a left contraction on the left side of the summand.
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	For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it’s nonzero.		For operands $$\mathbf a$$ and $$\mathbf b$$ having grades $$g$$ and $$h$$, the transwedge product of order $$k$$ generates a result having grade $$g + h - 2k$$, assuming it’s nonzero.

	The sum of all possible transwedge products yields the geometric product. That is,		The sum of all possible transwedge products, negating for orders 2 and 3 modulo 4, yields the geometric product. That is,

	:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,		:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A53}}} \mathbf b}$$,

	where $$n$$ is the dimension of the algebra.		where $$n$$ is the dimension of the algebra.
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	:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,		:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,

	where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$~~, and we have dropped the reverse operation because $$\mathbf a$$ is a vector~~.		where the [[dot product]] on the right is the result of the right contraction $$\mathbf b \vee \mathbf a^{\unicode{x2605}}$$.

	For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have		For bivectors $$\mathbf a$$ and $$\mathbf b$$, we have

	:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,		:$$\displaystyle\mathbf a \mathbin{\unicode{x27D1}} \mathbf b = \mathbf a \mathbin{\underset{0}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b - \mathbf a \mathbin{\underset{2}{\unicode{x2A53}}} \mathbf b = \mathbf a \wedge \mathbf b + \mathbf a \mathbin{\underset{1}{\unicode{x2A53}}} \mathbf b + \mathbf a \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf b$$,

	where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.		where the [[dot product]] is again attributable to the right contraction $$\mathbf b \vee \mathbf{\tilde a}^{\unicode{x2605}}$$. The middle term given by the transwedge product of order 1 generates the grade-2 part of the geometric product.
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	The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as		The ''transwedge antiproduct'' of order $$k$$ is the dual of the transwedge product of order $$k$$. It is written with a double downward wedge having either a subscript or an underscript indicating the order. It is defined as

	:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf~~{\smash{\mathbf{\underset{\Large\unicode{x7E}}{~~c~~}}}~~^{\unicode{x2606}}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf~~{\smash{\mathbf{\underset{\Large\unicode{x7E}}{c}}}_~~{\unicode{x2606}}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,		:$$\displaystyle\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf{\underline c} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf c^{\unicode{x2606}})} = \sum_{\mathbf c \in \mathcal B_{n - k}}{(\mathbf c_{\unicode{x2606}} \wedge \mathbf a) \vee (\mathbf b \wedge \mathbf{\overline c})}$$,

	where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual~~]], the [[reverse]] has become the [[antireverse~~]], and we are now summing over all basis elements of [[antigrade]] $$k$$.		where the wedge and antiwedge products have traded places, the [[bulk dual]] has become the [[weight dual]], and we are now summing over all basis elements of [[antigrade]] $$k$$.

	The sum of all possible transwedge antiproducts yields the geometric antiproduct. That is,		The sum of all possible transwedge antiproducts, again negating for orders 2 and 3 modulo 4, yields the geometric antiproduct. That is,

	:$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,		:$$\displaystyle\mathbf a \mathbin{\unicode{x27C7}} \mathbf b = \sum_{k=0}^n{(-1)^{k(k-1)/2}\,\mathbf a \mathbin{\underset{k}{\unicode{x2A54}}} \mathbf b}$$,

	The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, 3, and 4.		The components of the geometric antiproduct are shown in the following table, which color codes the transwedge antiproducts of order 0, 1, 2, 3, and 4.