Rank-3 scale theorems - Revision history

VectorGraphics at 00:05, 28 June 2025

2025-06-28T00:05:34Z

← Older revision		Revision as of 00:05, 28 June 2025
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	* Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)		* Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)
	* Triple Fokker blocks form a [http://en.wikipedia.org/wiki/Trihexagonal_tiling trihexagonal tiling] on the lattice.		* Triple Fokker blocks form a [http://en.wikipedia.org/wiki/Trihexagonal_tiling trihexagonal tiling] on the lattice.
	* A scale imprint is that of a Fokker block if and only if it is the [[product word]] of two DE scale imprints with the same number of notes. See [https://link.springer.com/chapter/10.1007/978-3-642-21590-2_24 Introduction to Scale Theory over Words in Two Dimensions \| SpringerLink]		* A scale imprint is that of a Fokker block if and only if it is the [[product word\|product]] of two DE scale imprints with the same number of notes. See [https://link.springer.com/chapter/10.1007/978-3-642-21590-2_24 Introduction to Scale Theory over Words in Two Dimensions \| SpringerLink]
	* If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L > m > n > s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s		* If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L > m > n > s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s
	* Any convex object on the lattice can be converted into a hexagon.		* Any convex object on the lattice can be converted into a hexagon.

Inthar at 22:25, 9 February 2024

2024-02-09T22:25:31Z

← Older revision		Revision as of 22:25, 9 February 2024
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	[[Category:Rank 3]]		[[Category:Rank 3]]
	[[Category:Scale]]		[[Category:Scale]]
	[[Category:~~Articles~~ with open problems]]		[[Category:Pages with open problems]]

Inthar: /* Unproven Conjectures */

2024-02-01T16:36:06Z

Unproven Conjectures

← Older revision		Revision as of 16:36, 1 February 2024
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	* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.		* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.
	* An MV3 scale always has two of the step sizes occurring the same number of times, except powers of abacaba. Except multi-period MV3's, such scales are always either pairwise-well-formed, a power of abcba, or a "twisted" word constructed from the mos 2qX rY. A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba. The PWF scales are exactly the single-period rank-3 [[billiard scales]].		* An MV3 scale always has two of the step sizes occurring the same number of times, except powers of abacaba. Except multi-period MV3's, such scales are always either pairwise-well-formed, a power of abcba, or a "twisted" word constructed from the mos 2qX rY. A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba. The PWF scales are exactly the single-period rank-3 [[billiard scales]].
	== ~~Unproven~~ Conjectures ==		== Conjectures ==
	* Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes.		* Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes.

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	[[Category:Rank 3]]		[[Category:Rank 3]]
	[[Category:Scale]]		[[Category:Scale]]
			[[Category:Articles with open problems]]

Inthar: Removed redirect to Ternary scale theorems

2023-07-31T18:15:52Z

Removed redirect to Ternary scale theorems

← Older revision		Revision as of 18:15, 31 July 2023
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	#~~redirect~~ [[~~Ternary scale theorems~~]]		== Theorems ==
			* Every triple [[Fokker block]] is max variety 3.
			* Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)
			* Triple Fokker blocks form a [http://en.wikipedia.org/wiki/Trihexagonal_tiling trihexagonal tiling] on the lattice.
			* A scale imprint is that of a Fokker block if and only if it is the [[product word]] of two DE scale imprints with the same number of notes. See [https://link.springer.com/chapter/10.1007/978-3-642-21590-2_24 Introduction to Scale Theory over Words in Two Dimensions \| SpringerLink]
			* If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L > m > n > s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s
			* Any convex object on the lattice can be converted into a hexagon.
			* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.
			* An MV3 scale always has two of the step sizes occurring the same number of times, except powers of abacaba. Except multi-period MV3's, such scales are always either pairwise-well-formed, a power of abcba, or a "twisted" word constructed from the mos 2qX rY. A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba. The PWF scales are exactly the single-period rank-3 [[billiard scales]].
			== Unproven Conjectures ==
			* Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes.

			[[Category:Fokker block]]
			[[Category:Math]]
			[[Category:Rank 3]]
			[[Category:Scale]]

Inthar: Also moving my proofs from "Generator-offset property".

2023-07-31T18:06:12Z

Also moving my proofs from "Generator-offset property".

← Older revision		Revision as of 18:06, 31 July 2023
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	~~== Theorems ==~~		#redirect [[Ternary scale theorems]]
	* Every triple [[Fokker block]] is max variety 3.
	* Every max variety 3 block is a triple Fokker block. (However, not every max-variety 3 scale, in general, need be a Fokker block.)
	* Triple Fokker blocks form a [http://en.wikipedia.org/wiki/Trihexagonal_tiling trihexagonal tiling] on the lattice.
	* A scale imprint is that of a Fokker block if and only if it is the [[~~product word]] of two DE scale imprints with the same number of notes. See~~ [~~https://link.springer.com/chapter/10.1007/978-3-642-21590-2_24 Introduction to Scale Theory over Words in Two Dimensions \| SpringerLink]~~
	* If the step sizes for a rank-3 Fokker block are L, m, n, and s, where L > m > n > s, then the following identity must hold: (n-s) + (m-s) = (L-s), hence n+m=L+s
	* Any convex object on the lattice can be converted into a hexagon.
	* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.
	* An MV3 scale always has two of the step sizes occurring the same number of times, except powers of abacaba. Except multi-period MV3's, such scales are always either pairwise-well-formed, a power of abcba, or a "twisted" word constructed from the mos 2qX rY. A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba. The PWF scales are exactly the single-period rank-3 [[billiard scales]].

	~~== Unproven Conjectures ==~~
	* Every rank-3 Fokker block has mean-variety < 4, meaning that some interval class will come in less than 4 sizes.

	~~[[Category:Fokker block]]~~
	~~[[Category:Math]]~~
	~~[[Category:Rank 3]]~~
	~~[[Category:Scale~~]]

Fredg999 category edits: Removing from Category:Theory using Cat-a-lot

2023-03-05T05:33:37Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 05:33, 5 March 2023
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	[[Category:Rank 3]]		[[Category:Rank 3]]
	[[Category:Scale]]		[[Category:Scale]]
	~~[[Category:Theory]]~~

Inthar: /* MV3 proofs */

2023-02-15T16:19:54Z

MV3 proofs

@@ Line 11: / Line 11: @@
 == Unproven Conjectures ==
 * Every rank-3 Fokker block has mean-variety &lt; 4, meaning that some interval class will come in less than 4 sizes.
 [[Category:Fokker block]]

Inthar: /* Theorems */

2023-02-15T16:19:19Z

Theorems

← Older revision		Revision as of 16:19, 15 February 2023
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	* Any convex object on the lattice can be converted into a hexagon.		* Any convex object on the lattice can be converted into a hexagon.
	* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.		* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.
	* An MV3 scale always has two of the step sizes occurring the same number of times, except powers of abacaba. Except multi-period MV3's, such scales are always either pairwise-well-formed, a power of abcba, or a "twisted" word constructed from the mos 2qX rY. A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba. The PWF scales are exactly the single-period rank-3 billiard scales.		* An MV3 scale always has two of the step sizes occurring the same number of times, except powers of abacaba. Except multi-period MV3's, such scales are always either pairwise-well-formed, a power of abcba, or a "twisted" word constructed from the mos 2qX rY. A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba. The PWF scales are exactly the single-period rank-3 [[billiard scales]].

	== Unproven Conjectures ==		== Unproven Conjectures ==

Inthar: /* Theorems */ This has been proven by Bulgakova et al. (2023)

2023-02-15T16:19:06Z

Theorems: This has been proven by Bulgakova et al. (2023)

← Older revision		Revision as of 16:19, 15 February 2023
Line 7:		Line 7:
	* Any convex object on the lattice can be converted into a hexagon.		* Any convex object on the lattice can be converted into a hexagon.
	* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.		* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.
	* A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba.		* An MV3 scale always has two of the step sizes occurring the same number of times, except powers of abacaba. Except multi-period MV3's, such scales are always either pairwise-well-formed, a power of abcba, or a "twisted" word constructed from the mos 2qX rY. A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba. The PWF scales are exactly the single-period rank-3 billiard scales.

	== Unproven Conjectures ==		== Unproven Conjectures ==

Inthar: /* Theorems */

2022-11-20T19:30:01Z

Theorems

← Older revision		Revision as of 19:30, 20 November 2022
Line 7:		Line 7:
	* Any convex object on the lattice can be converted into a hexagon.		* Any convex object on the lattice can be converted into a hexagon.
	* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.		* Any convex scale with 3 step sizes is a hexagon on the lattice, in which each set of parallel lines corresponds to one of the steps.
			* A pairwise-well-formed scale has odd size, and is either [[generator-offset]] or of the form abacaba.

	== Unproven Conjectures ==		== Unproven Conjectures ==