OEIS/FASS curves - Revision history

imported>Gfis: no stairs

2018-05-30T12:21:34Z

no stairs

← Older revision		Revision as of 12:21, 30 May 2018
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	====Elementary waves on the diagonal====		====Elementary waves on the diagonal====
	It appears that continuing paths must contain ''wave'' shapes at least on one of the diagonal nodes 22, 33 .. aa. The 3-wave has the "n" shape, the 5-wave looks like "NN", and in general the b-wave is 00~01..0b~1b~1a..10~20..2b..ba~bb. The shapes on the diagonal nodes will be rotated, but only in 2 out of 4 possible ways, depending on the parity of the row number.		It appears that continuing paths must contain ''wave'' shapes at least on one of the diagonal nodes 22, 33 .. aa. The 3-wave has the "n" shape, the 5-wave looks like "NN", and in general the b-wave is 00~01..0b~1b~1a..10~20..2b..ba~bb. The shapes on the diagonal nodes will be rotated, but only in 2 out of 4 possible ways, depending on the parity of the row number.
			====No "stairs"====
			It seems - at least for odd bases - that path segments which change the direction 4 times (or more often) in a row are not allowed:
			XX==XX XX==XX
			\|\| \|\|
			XX==XX or XX==XX
			\|\| \|\|
			XX XX
	====Implementation====		====Implementation====
	The Perl program <code>[https://github.com/gfis/fasces/blob/master/data/gen_paths.pl gen_paths.pl]</code> uses the restrictions mentioned above.		The Perl program <code>[https://github.com/gfis/fasces/blob/master/data/gen_paths.pl gen_paths.pl]</code> uses the restrictions mentioned above.

imported>Gfis: typo

2018-05-30T12:13:37Z

typo

← Older revision		Revision as of 12:13, 30 May 2018
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	/>[http://www.teherba.org/threejs/meander-4.html?base=5&title=&label=on&vec=Ln+0%2C1%2C2%2C3%2C4%2C14%2C13%2C12%2C11%2C10%2C20%2C30%2C40%2C41%2C31%2C21%2C22%2C23%2C24%2C34%2C33%2C32%2C42%2C43%2C44%2C144%2C143%2C142%2C132%2C133%2C134%2C124%2C123%2C122%2C121%2C131%2C141%2C140%2C130%2C120%2C110%2C111%2C112%2C113%2C114%2C104%2C103%2C102%2C101%2C100%2C200%2C300%2C400%2C401%2C301%2C201%2C202%2C203%2C204%2C304%2C303%2C302%2C402%2C403%2C404%2C414%2C413%2C412%2C312%2C313%2C314%2C214%2C213%2C212%2C211%2C311%2C411%2C410%2C310%2C210%2C220%2C230%2C240%2C340%2C330%2C320%2C420%2C430%2C440%2C441%2C431%2C421%2C321%2C331%2C341%2C241%2C231%2C221%2C222%2C223%2C224%2C234%2C233%2C232%2C242%2C243%2C244%2C344%2C343%2C342%2C332%2C333%2C334%2C324%2C323%2C322%2C422%2C423%2C424%2C434%2C433%2C432%2C442%2C443%2C444%2C444 Ln] - asymmetric, similiar to Kn <br		/>[http://www.teherba.org/threejs/meander-4.html?base=5&title=&label=on&vec=Ln+0%2C1%2C2%2C3%2C4%2C14%2C13%2C12%2C11%2C10%2C20%2C30%2C40%2C41%2C31%2C21%2C22%2C23%2C24%2C34%2C33%2C32%2C42%2C43%2C44%2C144%2C143%2C142%2C132%2C133%2C134%2C124%2C123%2C122%2C121%2C131%2C141%2C140%2C130%2C120%2C110%2C111%2C112%2C113%2C114%2C104%2C103%2C102%2C101%2C100%2C200%2C300%2C400%2C401%2C301%2C201%2C202%2C203%2C204%2C304%2C303%2C302%2C402%2C403%2C404%2C414%2C413%2C412%2C312%2C313%2C314%2C214%2C213%2C212%2C211%2C311%2C411%2C410%2C310%2C210%2C220%2C230%2C240%2C340%2C330%2C320%2C420%2C430%2C440%2C441%2C431%2C421%2C321%2C331%2C341%2C241%2C231%2C221%2C222%2C223%2C224%2C234%2C233%2C232%2C242%2C243%2C244%2C344%2C343%2C342%2C332%2C333%2C334%2C324%2C323%2C322%2C422%2C423%2C424%2C434%2C433%2C432%2C442%2C443%2C444%2C444 Ln] - asymmetric, similiar to Kn <br
	/>[http://www.teherba.org/threejs/meander-4.html?base=5&title=&label=on&decout=on&vec=Ls+0%2C1%2C2%2C3%2C4%2C14%2C13%2C12%2C11%2C10%2C20%2C30%2C40%2C41%2C31%2C21%2C22%2C32%2C42%2C43%2C33%2C23%2C24%2C34%2C44%0D%0A%2C144%2C134%2C124%2C123%2C133%2C143%2C142%2C132%2C122%2C121%2C131%2C141%2C140%2C130%2C120%2C110%2C111%2C112%2C113%2C114%0D%0A%2C104%2C103%2C102%2C101%2C100%2C200%2C300%2C400%2C401%2C301%2C201%2C202%2C302%2C402%2C403%2C303%2C203%2C204%2C304%2C404%0D%0A%2C414%2C314%2C214%2C213%2C313%2C413%2C412%2C312%2C212%2C211%2C311%2C411%2C410%2C310%2C210%2C220%2C320%2C420%2C430%2C330%0D%0A%2C230%2C240%2C340%2C440%2C441%2C341%2C241%2C231%2C331%2C431%2C421%2C321%2C221%2C222%2C322%2C422%2C432%2C332%2C232%2C242%0D%0A%2C342%2C442%2C443%2C343%2C243%2C233%2C333%2C433%2C423%2C323%2C223%2C224%2C324%2C424%2C434%2C334%2C234%2C244%2C344%2C444%0D%0A%2C444 Ls] - asymmetric, similiar to Fs<br		/>[http://www.teherba.org/threejs/meander-4.html?base=5&title=&label=on&decout=on&vec=Ls+0%2C1%2C2%2C3%2C4%2C14%2C13%2C12%2C11%2C10%2C20%2C30%2C40%2C41%2C31%2C21%2C22%2C32%2C42%2C43%2C33%2C23%2C24%2C34%2C44%0D%0A%2C144%2C134%2C124%2C123%2C133%2C143%2C142%2C132%2C122%2C121%2C131%2C141%2C140%2C130%2C120%2C110%2C111%2C112%2C113%2C114%0D%0A%2C104%2C103%2C102%2C101%2C100%2C200%2C300%2C400%2C401%2C301%2C201%2C202%2C302%2C402%2C403%2C303%2C203%2C204%2C304%2C404%0D%0A%2C414%2C314%2C214%2C213%2C313%2C413%2C412%2C312%2C212%2C211%2C311%2C411%2C410%2C310%2C210%2C220%2C320%2C420%2C430%2C330%0D%0A%2C230%2C240%2C340%2C440%2C441%2C341%2C241%2C231%2C331%2C431%2C421%2C321%2C221%2C222%2C322%2C422%2C432%2C332%2C232%2C242%0D%0A%2C342%2C442%2C443%2C343%2C243%2C233%2C333%2C433%2C423%2C323%2C223%2C224%2C324%2C424%2C434%2C334%2C234%2C244%2C344%2C444%0D%0A%2C444 Ls] - asymmetric, similiar to Fs<br
	/>[http://www.teherba.org/threejs/meander-4.html?base=5&title=&label=on&vec=NN+%5B0%2C1%2C2%2C3%2C4%2C14%2C13%2C12%2C11%2C10%2C20%2C21%2C22%2C23%2C24%2C34%2C33%2C32%2C31%2C30%2C40%2C41%2C42%2C43%2C44%2C144%2C143%0D%0A%2C142%2C141%2C140%2C130%2C131%2C132%2C133%2C134%2C124%2C123%2C122%2C121%2C120%2C110%2C111%2C112%2C113%2C114%2C104%2C103%2C102%2C101%2C100%2C200%2C201%2C202%2C203%2C204%2C214%2C213%2C212%2C211%2C210%2C220%2C221%2C222%2C223%2C224%2C234%2C233%0D%0A%2C232%2C231%2C230%2C240%2C241%2C242%2C243%2C244%2C344%2C343%2C342%2C341%2C340%2C330%2C331%2C332%2C333%2C334%2C324%2C323%2C322%0D%0A%2C321%2C320%2C310%2C311%2C312%2C313%2C314%2C304%2C303%2C302%2C301%2C300%2C400%2C401%2C402%2C403%2C404%2C414%2C413%0D%0A%2C412%2C411%2C410%2C420%2C421%2C422%2C423%2C424%2C434%2C433%2C432%2C431%2C430%2C440%2C441%2C442%2C443%2C444%2C1444%2C%5D NN]] - simple wave		/>[http://www.teherba.org/threejs/meander-4.html?base=5&title=&label=on&vec=NN+%5B0%2C1%2C2%2C3%2C4%2C14%2C13%2C12%2C11%2C10%2C20%2C21%2C22%2C23%2C24%2C34%2C33%2C32%2C31%2C30%2C40%2C41%2C42%2C43%2C44%2C144%2C143%0D%0A%2C142%2C141%2C140%2C130%2C131%2C132%2C133%2C134%2C124%2C123%2C122%2C121%2C120%2C110%2C111%2C112%2C113%2C114%2C104%2C103%2C102%2C101%2C100%2C200%2C201%2C202%2C203%2C204%2C214%2C213%2C212%2C211%2C210%2C220%2C221%2C222%2C223%2C224%2C234%2C233%0D%0A%2C232%2C231%2C230%2C240%2C241%2C242%2C243%2C244%2C344%2C343%2C342%2C341%2C340%2C330%2C331%2C332%2C333%2C334%2C324%2C323%2C322%0D%0A%2C321%2C320%2C310%2C311%2C312%2C313%2C314%2C304%2C303%2C302%2C301%2C300%2C400%2C401%2C402%2C403%2C404%2C414%2C413%0D%0A%2C412%2C411%2C410%2C420%2C421%2C422%2C423%2C424%2C434%2C433%2C432%2C431%2C430%2C440%2C441%2C442%2C443%2C444%2C1444%2C%5D NN] - simple wave
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	\| 7		\| 7
	\| [http://www.teherba.org/threejs/meander-4.html?base=7&title=&label=on&vec=%5B0%2C1%2C2%2C3%2C4%2C5%2C6%2C16%2C26%2C36%2C46%2C56%2C55%2C54%2C53%2C52%2C51%2C%0D%0A41%2C31%2C21%2C22%2C23%2C24%2C34%2C33%2C32%2C42%2C43%2C44%2C45%2C35%2C25%2C15%2C%0D%0A14%2C13%2C12%2C11%2C10%2C20%2C30%2C40%2C50%2C60%2C61%2C62%2C63%2C64%2C65%2C66%2C%0D%0A166%2C266%2C366%2C466%2C566%2C565%2C564%2C563%2C562%2C561%2C461%2C361%2C261%2C262%2C263%2C264%2C%0D%0A364%2C363%2C362%2C462%2C463%2C464%2C465%2C365%2C265%2C165%2C164%2C163%2C162%2C161%2C160%2C260%2C%0D%0A360%2C460%2C560%2C550%2C540%2C530%2C520%2C510%2C410%2C310%2C210%2C220%2C230%2C240%2C340%2C330%2C%0D%0A320%2C420%2C430%2C440%2C450%2C350%2C250%2C150%2C140%2C130%2C120%2C110%2C111%2C112%2C113%2C114%2C%0D%0A115%2C125%2C135%2C145%2C144%2C143%2C142%2C132%2C133%2C134%2C124%2C123%2C122%2C121%2C131%2C141%2C%0D%0A151%2C152%2C153%2C154%2C155%2C255%2C355%2C455%2C454%2C453%2C452%2C352%2C353%2C354%2C254%2C253%2C%0D%0A252%2C251%2C351%2C451%2C441%2C431%2C421%2C321%2C331%2C341%2C241%2C231%2C221%2C222%2C223%2C224%2C%0D%0A234%2C233%2C232%2C242%2C243%2C244%2C344%2C343%2C342%2C332%2C333%2C334%2C324%2C323%2C322%2C422%2C%0D%0A423%2C424%2C434%2C433%2C432%2C442%2C443%2C444%2C445%2C435%2C425%2C325%2C335%2C345%2C245%2C235%2C%0D%0A225%2C215%2C315%2C415%2C414%2C413%2C412%2C312%2C313%2C314%2C214%2C213%2C212%2C211%2C311%2C411%2C%0D%0A511%2C512%2C513%2C514%2C515%2C525%2C535%2C545%2C544%2C543%2C542%2C532%2C533%2C534%2C524%2C523%2C%0D%0A522%2C521%2C531%2C541%2C551%2C552%2C553%2C554%2C555%2C556%2C546%2C536%2C526%2C516%2C416%2C316%2C%0D%0A216%2C226%2C236%2C246%2C346%2C336%2C326%2C426%2C436%2C446%2C456%2C356%2C256%2C156%2C146%2C136%2C%0D%0A126%2C116%2C106%2C206%2C306%2C406%2C506%2C505%2C504%2C503%2C502%2C501%2C401%2C301%2C201%2C202%2C%0D%0A203%2C204%2C304%2C303%2C302%2C402%2C403%2C404%2C405%2C305%2C205%2C105%2C104%2C103%2C102%2C101%2C%0D%0A100%2C200%2C300%2C400%2C500%2C600%2C601%2C602%2C603%2C604%2C605%2C606%2C616%2C626%2C636%2C646%2C%0D%0A656%2C655%2C654%2C653%2C652%2C651%2C641%2C631%2C621%2C622%2C623%2C624%2C634%2C633%2C632%2C642%2C%0D%0A643%2C644%2C645%2C635%2C625%2C615%2C614%2C613%2C612%2C611%2C610%2C620%2C630%2C640%2C650%2C660%2C%0D%0A661%2C662%2C663%2C664%2C665%2C666%2C1666%2C%5D Kn7] - like Kn, c.f. [https://oeis.org/A300857 A300857]<br />http://www.teherba.org/threejs/meander-4.html?base=7&title=&vec=%5B0%2C1%2C2%2C3%2C4%2C5%2C6%2C16%2C15%2C14%2C13%2C12%2C11%2C10%2C20%2C30%2C40%2C%0D%0A41%2C31%2C21%2C22%2C32%2C42%2C43%2C33%2C23%2C24%2C34%2C44%2C45%2C35%2C25%2C26%2C%0D%0A36%2C46%2C56%2C55%2C54%2C53%2C52%2C51%2C50%2C60%2C61%2C62%2C63%2C64%2C65%2C66%2C%0D%0A166%2C165%2C164%2C163%2C162%2C161%2C160%2C150%2C151%2C152%2C153%2C154%2C155%2C156%2C146%2C136%2C%0D%0A126%2C125%2C135%2C145%2C144%2C134%2C124%2C123%2C133%2C143%2C142%2C132%2C122%2C121%2C131%2C141%2C%0D%0A140%2C130%2C120%2C110%2C111%2C112%2C113%2C114%2C115%2C116%2C106%2C105%2C104%2C103%2C102%2C101%2C%0D%0A100%2C200%2C300%2C400%2C401%2C301%2C201%2C202%2C302%2C402%2C403%2C303%2C203%2C204%2C304%2C404%2C%0D%0A405%2C305%2C205%2C206%2C306%2C406%2C416%2C316%2C216%2C215%2C315%2C415%2C414%2C314%2C214%2C213%2C%0D%0A313%2C413%2C412%2C312%2C212%2C211%2C311%2C411%2C410%2C310%2C210%2C220%2C320%2C420%2C430%2C330%2C%0D%0A230%2C240%2C340%2C440%2C441%2C341%2C241%2C231%2C331%2C431%2C421%2C321%2C221%2C222%2C322%2C422%2C%0D%0A432%2C332%2C232%2C242%2C342%2C442%2C443%2C343%2C243%2C233%2C333%2C433%2C423%2C323%2C223%2C224%2C%0D%0A324%2C424%2C434%2C334%2C234%2C244%2C344%2C444%2C445%2C345%2C245%2C235%2C335%2C435%2C425%2C325%2C%0D%0A225%2C226%2C326%2C426%2C436%2C336%2C236%2C246%2C346%2C446%2C456%2C356%2C256%2C255%2C355%2C455%2C%0D%0A454%2C354%2C254%2C253%2C353%2C453%2C452%2C352%2C252%2C251%2C351%2C451%2C450%2C350%2C250%2C260%2C%0D%0A360%2C460%2C461%2C361%2C261%2C262%2C362%2C462%2C463%2C363%2C263%2C264%2C364%2C464%2C465%2C365%2C%0D%0A265%2C266%2C366%2C466%2C566%2C565%2C564%2C563%2C562%2C561%2C560%2C550%2C551%2C552%2C553%2C554%2C%0D%0A555%2C556%2C546%2C536%2C526%2C525%2C535%2C545%2C544%2C534%2C524%2C523%2C533%2C543%2C542%2C532%2C%0D%0A522%2C521%2C531%2C541%2C540%2C530%2C520%2C510%2C511%2C512%2C513%2C514%2C515%2C516%2C506%2C505%2C%0D%0A504%2C503%2C502%2C501%2C500%2C600%2C601%2C602%2C603%2C604%2C605%2C606%2C616%2C615%2C614%2C613%2C%0D%0A612%2C611%2C610%2C620%2C630%2C640%2C641%2C631%2C621%2C622%2C632%2C642%2C643%2C633%2C623%2C624%2C%0D%0A634%2C644%2C645%2C635%2C625%2C626%2C636%2C646%2C656%2C655%2C654%2C653%2C652%2C651%2C650%2C660%2C%0D%0A661%2C662%2C663%2C664%2C665%2C666%2C1666%2C%5D IssI], another shape<br		\| [http://www.teherba.org/threejs/meander-4.html?base=7&title=&label=on&vec=%5B0%2C1%2C2%2C3%2C4%2C5%2C6%2C16%2C26%2C36%2C46%2C56%2C55%2C54%2C53%2C52%2C51%2C%0D%0A41%2C31%2C21%2C22%2C23%2C24%2C34%2C33%2C32%2C42%2C43%2C44%2C45%2C35%2C25%2C15%2C%0D%0A14%2C13%2C12%2C11%2C10%2C20%2C30%2C40%2C50%2C60%2C61%2C62%2C63%2C64%2C65%2C66%2C%0D%0A166%2C266%2C366%2C466%2C566%2C565%2C564%2C563%2C562%2C561%2C461%2C361%2C261%2C262%2C263%2C264%2C%0D%0A364%2C363%2C362%2C462%2C463%2C464%2C465%2C365%2C265%2C165%2C164%2C163%2C162%2C161%2C160%2C260%2C%0D%0A360%2C460%2C560%2C550%2C540%2C530%2C520%2C510%2C410%2C310%2C210%2C220%2C230%2C240%2C340%2C330%2C%0D%0A320%2C420%2C430%2C440%2C450%2C350%2C250%2C150%2C140%2C130%2C120%2C110%2C111%2C112%2C113%2C114%2C%0D%0A115%2C125%2C135%2C145%2C144%2C143%2C142%2C132%2C133%2C134%2C124%2C123%2C122%2C121%2C131%2C141%2C%0D%0A151%2C152%2C153%2C154%2C155%2C255%2C355%2C455%2C454%2C453%2C452%2C352%2C353%2C354%2C254%2C253%2C%0D%0A252%2C251%2C351%2C451%2C441%2C431%2C421%2C321%2C331%2C341%2C241%2C231%2C221%2C222%2C223%2C224%2C%0D%0A234%2C233%2C232%2C242%2C243%2C244%2C344%2C343%2C342%2C332%2C333%2C334%2C324%2C323%2C322%2C422%2C%0D%0A423%2C424%2C434%2C433%2C432%2C442%2C443%2C444%2C445%2C435%2C425%2C325%2C335%2C345%2C245%2C235%2C%0D%0A225%2C215%2C315%2C415%2C414%2C413%2C412%2C312%2C313%2C314%2C214%2C213%2C212%2C211%2C311%2C411%2C%0D%0A511%2C512%2C513%2C514%2C515%2C525%2C535%2C545%2C544%2C543%2C542%2C532%2C533%2C534%2C524%2C523%2C%0D%0A522%2C521%2C531%2C541%2C551%2C552%2C553%2C554%2C555%2C556%2C546%2C536%2C526%2C516%2C416%2C316%2C%0D%0A216%2C226%2C236%2C246%2C346%2C336%2C326%2C426%2C436%2C446%2C456%2C356%2C256%2C156%2C146%2C136%2C%0D%0A126%2C116%2C106%2C206%2C306%2C406%2C506%2C505%2C504%2C503%2C502%2C501%2C401%2C301%2C201%2C202%2C%0D%0A203%2C204%2C304%2C303%2C302%2C402%2C403%2C404%2C405%2C305%2C205%2C105%2C104%2C103%2C102%2C101%2C%0D%0A100%2C200%2C300%2C400%2C500%2C600%2C601%2C602%2C603%2C604%2C605%2C606%2C616%2C626%2C636%2C646%2C%0D%0A656%2C655%2C654%2C653%2C652%2C651%2C641%2C631%2C621%2C622%2C623%2C624%2C634%2C633%2C632%2C642%2C%0D%0A643%2C644%2C645%2C635%2C625%2C615%2C614%2C613%2C612%2C611%2C610%2C620%2C630%2C640%2C650%2C660%2C%0D%0A661%2C662%2C663%2C664%2C665%2C666%2C1666%2C%5D Kn7] - like Kn, c.f. [https://oeis.org/A300857 A300857]<br />[http://www.teherba.org/threejs/meander-4.html?base=7&title=&vec=%5B0%2C1%2C2%2C3%2C4%2C5%2C6%2C16%2C15%2C14%2C13%2C12%2C11%2C10%2C20%2C30%2C40%2C%0D%0A41%2C31%2C21%2C22%2C32%2C42%2C43%2C33%2C23%2C24%2C34%2C44%2C45%2C35%2C25%2C26%2C%0D%0A36%2C46%2C56%2C55%2C54%2C53%2C52%2C51%2C50%2C60%2C61%2C62%2C63%2C64%2C65%2C66%2C%0D%0A166%2C165%2C164%2C163%2C162%2C161%2C160%2C150%2C151%2C152%2C153%2C154%2C155%2C156%2C146%2C136%2C%0D%0A126%2C125%2C135%2C145%2C144%2C134%2C124%2C123%2C133%2C143%2C142%2C132%2C122%2C121%2C131%2C141%2C%0D%0A140%2C130%2C120%2C110%2C111%2C112%2C113%2C114%2C115%2C116%2C106%2C105%2C104%2C103%2C102%2C101%2C%0D%0A100%2C200%2C300%2C400%2C401%2C301%2C201%2C202%2C302%2C402%2C403%2C303%2C203%2C204%2C304%2C404%2C%0D%0A405%2C305%2C205%2C206%2C306%2C406%2C416%2C316%2C216%2C215%2C315%2C415%2C414%2C314%2C214%2C213%2C%0D%0A313%2C413%2C412%2C312%2C212%2C211%2C311%2C411%2C410%2C310%2C210%2C220%2C320%2C420%2C430%2C330%2C%0D%0A230%2C240%2C340%2C440%2C441%2C341%2C241%2C231%2C331%2C431%2C421%2C321%2C221%2C222%2C322%2C422%2C%0D%0A432%2C332%2C232%2C242%2C342%2C442%2C443%2C343%2C243%2C233%2C333%2C433%2C423%2C323%2C223%2C224%2C%0D%0A324%2C424%2C434%2C334%2C234%2C244%2C344%2C444%2C445%2C345%2C245%2C235%2C335%2C435%2C425%2C325%2C%0D%0A225%2C226%2C326%2C426%2C436%2C336%2C236%2C246%2C346%2C446%2C456%2C356%2C256%2C255%2C355%2C455%2C%0D%0A454%2C354%2C254%2C253%2C353%2C453%2C452%2C352%2C252%2C251%2C351%2C451%2C450%2C350%2C250%2C260%2C%0D%0A360%2C460%2C461%2C361%2C261%2C262%2C362%2C462%2C463%2C363%2C263%2C264%2C364%2C464%2C465%2C365%2C%0D%0A265%2C266%2C366%2C466%2C566%2C565%2C564%2C563%2C562%2C561%2C560%2C550%2C551%2C552%2C553%2C554%2C%0D%0A555%2C556%2C546%2C536%2C526%2C525%2C535%2C545%2C544%2C534%2C524%2C523%2C533%2C543%2C542%2C532%2C%0D%0A522%2C521%2C531%2C541%2C540%2C530%2C520%2C510%2C511%2C512%2C513%2C514%2C515%2C516%2C506%2C505%2C%0D%0A504%2C503%2C502%2C501%2C500%2C600%2C601%2C602%2C603%2C604%2C605%2C606%2C616%2C615%2C614%2C613%2C%0D%0A612%2C611%2C610%2C620%2C630%2C640%2C641%2C631%2C621%2C622%2C632%2C642%2C643%2C633%2C623%2C624%2C%0D%0A634%2C644%2C645%2C635%2C625%2C626%2C636%2C646%2C656%2C655%2C654%2C653%2C652%2C651%2C650%2C660%2C%0D%0A661%2C662%2C663%2C664%2C665%2C666%2C1666%2C%5D IssI], another shape<br
	/>[http://www.teherba.org/threejs/meander-4.html?base=7&title=&vec=%5B0%2C1%2C2%2C3%2C4%2C5%2C6%2C16%2C15%2C14%2C13%2C12%2C11%2C10%2C20%2C30%2C40%2C%0D%0A41%2C31%2C21%2C22%2C23%2C24%2C34%2C33%2C32%2C42%2C43%2C44%2C45%2C35%2C25%2C26%2C%0D%0A36%2C46%2C56%2C55%2C54%2C53%2C52%2C51%2C50%2C60%2C61%2C62%2C63%2C64%2C65%2C66%2C%0D%0A166%2C165%2C164%2C163%2C162%2C161%2C160%2C150%2C151%2C152%2C153%2C154%2C155%2C156%2C146%2C136%2C%0D%0A126%2C125%2C135%2C145%2C144%2C143%2C142%2C132%2C133%2C134%2C124%2C123%2C122%2C121%2C131%2C141%2C%0D%0A140%2C130%2C120%2C110%2C111%2C112%2C113%2C114%2C115%2C116%2C106%2C105%2C104%2C103%2C102%2C101%2C%0D%0A100%2C200%2C300%2C400%2C401%2C301%2C201%2C202%2C203%2C204%2C304%2C303%2C302%2C402%2C403%2C404%2C%0D%0A405%2C305%2C205%2C206%2C306%2C406%2C416%2C316%2C216%2C215%2C315%2C415%2C414%2C413%2C412%2C312%2C%0D%0A313%2C314%2C214%2C213%2C212%2C211%2C311%2C411%2C410%2C310%2C210%2C220%2C230%2C240%2C340%2C330%2C%0D%0A320%2C420%2C430%2C440%2C441%2C431%2C421%2C321%2C331%2C341%2C241%2C231%2C221%2C222%2C223%2C224%2C%0D%0A234%2C233%2C232%2C242%2C243%2C244%2C344%2C343%2C342%2C332%2C333%2C334%2C324%2C323%2C322%2C422%2C%0D%0A423%2C424%2C434%2C433%2C432%2C442%2C443%2C444%2C445%2C435%2C425%2C325%2C335%2C345%2C245%2C235%2C%0D%0A225%2C226%2C236%2C246%2C346%2C336%2C326%2C426%2C436%2C446%2C456%2C356%2C256%2C255%2C355%2C455%2C%0D%0A454%2C453%2C452%2C352%2C353%2C354%2C254%2C253%2C252%2C251%2C351%2C451%2C450%2C350%2C250%2C260%2C%0D%0A360%2C460%2C461%2C361%2C261%2C262%2C263%2C264%2C364%2C363%2C362%2C462%2C463%2C464%2C465%2C365%2C%0D%0A265%2C266%2C366%2C466%2C566%2C565%2C564%2C563%2C562%2C561%2C560%2C550%2C551%2C552%2C553%2C554%2C%0D%0A555%2C556%2C546%2C536%2C526%2C525%2C535%2C545%2C544%2C543%2C542%2C532%2C533%2C534%2C524%2C523%2C%0D%0A522%2C521%2C531%2C541%2C540%2C530%2C520%2C510%2C511%2C512%2C513%2C514%2C515%2C516%2C506%2C505%2C%0D%0A504%2C503%2C502%2C501%2C500%2C600%2C601%2C602%2C603%2C604%2C605%2C606%2C616%2C615%2C614%2C613%2C%0D%0A612%2C611%2C610%2C620%2C630%2C640%2C641%2C631%2C621%2C622%2C623%2C624%2C634%2C633%2C632%2C642%2C%0D%0A643%2C644%2C645%2C635%2C625%2C626%2C636%2C646%2C656%2C655%2C654%2C653%2C652%2C651%2C650%2C660%2C%0D%0A661%2C662%2C663%2C664%2C665%2C666%2C1666%2C%5D LnT], yet another		/>[http://www.teherba.org/threejs/meander-4.html?base=7&title=&vec=%5B0%2C1%2C2%2C3%2C4%2C5%2C6%2C16%2C15%2C14%2C13%2C12%2C11%2C10%2C20%2C30%2C40%2C%0D%0A41%2C31%2C21%2C22%2C23%2C24%2C34%2C33%2C32%2C42%2C43%2C44%2C45%2C35%2C25%2C26%2C%0D%0A36%2C46%2C56%2C55%2C54%2C53%2C52%2C51%2C50%2C60%2C61%2C62%2C63%2C64%2C65%2C66%2C%0D%0A166%2C165%2C164%2C163%2C162%2C161%2C160%2C150%2C151%2C152%2C153%2C154%2C155%2C156%2C146%2C136%2C%0D%0A126%2C125%2C135%2C145%2C144%2C143%2C142%2C132%2C133%2C134%2C124%2C123%2C122%2C121%2C131%2C141%2C%0D%0A140%2C130%2C120%2C110%2C111%2C112%2C113%2C114%2C115%2C116%2C106%2C105%2C104%2C103%2C102%2C101%2C%0D%0A100%2C200%2C300%2C400%2C401%2C301%2C201%2C202%2C203%2C204%2C304%2C303%2C302%2C402%2C403%2C404%2C%0D%0A405%2C305%2C205%2C206%2C306%2C406%2C416%2C316%2C216%2C215%2C315%2C415%2C414%2C413%2C412%2C312%2C%0D%0A313%2C314%2C214%2C213%2C212%2C211%2C311%2C411%2C410%2C310%2C210%2C220%2C230%2C240%2C340%2C330%2C%0D%0A320%2C420%2C430%2C440%2C441%2C431%2C421%2C321%2C331%2C341%2C241%2C231%2C221%2C222%2C223%2C224%2C%0D%0A234%2C233%2C232%2C242%2C243%2C244%2C344%2C343%2C342%2C332%2C333%2C334%2C324%2C323%2C322%2C422%2C%0D%0A423%2C424%2C434%2C433%2C432%2C442%2C443%2C444%2C445%2C435%2C425%2C325%2C335%2C345%2C245%2C235%2C%0D%0A225%2C226%2C236%2C246%2C346%2C336%2C326%2C426%2C436%2C446%2C456%2C356%2C256%2C255%2C355%2C455%2C%0D%0A454%2C453%2C452%2C352%2C353%2C354%2C254%2C253%2C252%2C251%2C351%2C451%2C450%2C350%2C250%2C260%2C%0D%0A360%2C460%2C461%2C361%2C261%2C262%2C263%2C264%2C364%2C363%2C362%2C462%2C463%2C464%2C465%2C365%2C%0D%0A265%2C266%2C366%2C466%2C566%2C565%2C564%2C563%2C562%2C561%2C560%2C550%2C551%2C552%2C553%2C554%2C%0D%0A555%2C556%2C546%2C536%2C526%2C525%2C535%2C545%2C544%2C543%2C542%2C532%2C533%2C534%2C524%2C523%2C%0D%0A522%2C521%2C531%2C541%2C540%2C530%2C520%2C510%2C511%2C512%2C513%2C514%2C515%2C516%2C506%2C505%2C%0D%0A504%2C503%2C502%2C501%2C500%2C600%2C601%2C602%2C603%2C604%2C605%2C606%2C616%2C615%2C614%2C613%2C%0D%0A612%2C611%2C610%2C620%2C630%2C640%2C641%2C631%2C621%2C622%2C623%2C624%2C634%2C633%2C632%2C642%2C%0D%0A643%2C644%2C645%2C635%2C625%2C626%2C636%2C646%2C656%2C655%2C654%2C653%2C652%2C651%2C650%2C660%2C%0D%0A661%2C662%2C663%2C664%2C665%2C666%2C1666%2C%5D LnT], yet another
	\|-		\|-

imported>Gfis: Examples

2018-05-30T12:11:09Z

Examples

@@ Line 71: / Line 71: @@
   ...    ?
 The basic building block (for odd bases) seems to be the last matrix for base 3 above. Only this one generates a path (up to the 6th dimension).
 ===3D Manufacturing===
 [[File:trinckle-box.jpg|250px|right|4x4 cm cube (polyamide sintering)]]

imported>Gfis: Path expressions

2018-05-28T09:01:12Z

Path expressions

← Older revision		Revision as of 09:01, 28 May 2018
Line 124:		Line 124:
	====Implementation====		====Implementation====
	The Perl program <code>[https://github.com/gfis/fasces/blob/master/data/gen_paths.pl gen_paths.pl]</code> uses the restrictions mentioned above.		The Perl program <code>[https://github.com/gfis/fasces/blob/master/data/gen_paths.pl gen_paths.pl]</code> uses the restrictions mentioned above.
			===Path expressions===
			In order to prove the desired properties, we will use a language which describes the paths up to some generation (dimension). We use the following operations to build path expressions:
			* A0 = [] : the empty path
			* X = [a,b,c] : build a path expression ''X'' from elements ''a,b,c''
			* "/X" : reverse the sequence of elements in path expression ''X''
			* "X.n+m" : insert digit ''m'' after the digit for ''base^n'' in each element of ''X''
			Examples:
			[].0+1 = [1];
			[12].0+3 = [123];
			[12].2+3 = [312];
			[1234].1+0 = [12304];
			With this notation, we can describe the paths for base 3 and base 5 in a rather consistent way. It should be obvious how additional expressions must be added for higher dimensions. Similiar shapes show very similiar path expressions.
			====Base 3====
			n A0 = [];
			A1 = [A0.0+0,/A0.0+1,A0.0+2];
			A2 = [A1.1+0,/A1.1+1,A1.1+2];
			A3 = [A2.2+0,/A2.2+1,A2.2+2];
			...
			s A0 = [];
			A1 = [A0.0+0,/A0.0+1,A0.0+2];
			A2 = [A1.0+0,/A1.0+1,A1.0+2];
			A3 = [A2.0+0,/A2.0+1,A2.0+2];
			...
			====Base 5====
			Fs A0 = [];
			A1 = [A0.0+3,/A0.0+2,A0.0+1];
			B1 = [A0.0+0,/A1,A0.0+4];
			A2 = [A1.0+3,/A1.0+2,A1.0+1];
			B2 = [A1.0+0,/A2,A1.0+4];
			C2 = [B1.1+0,/B2,B1.1+4];
			A3 = [A2.0+3,/A2.0+2,A2.0+1];
			B3 = [A2.0+0,/A3,A2.0+4];
			C3 = [B2.1+0,/B3,B2.1+4];
			D3 = [C2.2+0,/C3,C2.2+4];
			...
			Kn A0 = [];
			A1 = [A0.0+3,/A0.0+2,A0.0+1];
			B1 = [A0.0+0,/A1,A0.0+4];
			A2 = [A1.1+3,/A1.1+2,A1.1+1];
			B2 = [A1.0+0,/A2,A1.0+4];
			C2 = [B1.1+0,/B2,B1.1+4]
			A3 = [A2.2+3,/A2.2+2,A2.2+1];
			B3 = [A2.0+0,/A3,A2.0+4];
			C3 = [B2.1+0,/B3,B2.1+4];
			D3 = [C2.2+0,/C3,C2.2+4];
			...
			Ln A0 = [];
			A1 = [A0.0+2,/A0.0+3,A0.0+4];
			B1 = [A0.0+0,/A0.0+1,A1];
			A2 = [A1.1+2,/A1.1+3,A1.1+4];
			B2 = [A1.0+0,/A1.0+1,A2];
			C2 = [B1.1+0,/B1.1+1,B2];
			A3 = [A2.2+2,/A2.2+3,A2.2+4];
			B3 = [A2.0+0,/A2.0+1,A3];
			C3 = [B2.1+0,/B2.1+1,B3];
			D3 = [C2.2+0,/C2.2+1,C3];
			...
			Ls A0 = [];
			A1 = [A0.0+2,/A0.0+3,A0.0+4];
			B1 = [A0.0+0,/A0.0+1,A1];
			A2 = [A1.0+2,/A1.0+3,A1.0+4];
			B2 = [A1.0+0,/A1.0+1,A2];
			C2 = [B1.1+0,/B1.1+1,B2];
			A3 = [A2.0+2,/A2.0+3,A2.0+4];
			B3 = [A2.0+0,/A2.0+1,A3];
			C3 = [B2.1+0,/B2.1+1,B3];
			D3 = [C2.2+0,/C2.2+1,C3];
			...
			NN A0 = [];
			A1 = [A0.0+0,/A0.0+1,A0.0+2,/A0.0+3,A0.0+4]
			A2 = [A1.1+0,/A1.1+1,A1.1+2,/A1.1+3,A1.1+4]
			A3 = [A2.2+0,/A2.2+1,A2.2+2,/A2.2+3,A2.2+4]
			...

imported>Gfis: wave shapes

2018-05-23T18:38:50Z

wave shapes

← Older revision		Revision as of 18:38, 23 May 2018
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	\|\| \|\|		\|\| \|\|
	00 10		00 10
			n
	3 4 02==12==22 02==12==22 02==12==22 02==12 22		3 4 02==12==22 02==12==22 02==12==22 02==12 22
	\|\| \|\| \|\| \|\| \|\| \|\| \|\| \|\|		\|\| \|\| \|\| \|\| \|\| \|\| \|\| \|\|
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	* If the path hits a border, one of the 2 neighbouring nodes must already be contained in the path.		* If the path hits a border, one of the 2 neighbouring nodes must already be contained in the path.
	If both neighbours were not visited so far, the path would divide the square in 2 halves, It would turn into one of them, and could never reach the non-visited nodes in the other one.		If both neighbours were not visited so far, the path would divide the square in 2 halves, It would turn into one of them, and could never reach the non-visited nodes in the other one.
			====Elementary waves on the diagonal====
			It appears that continuing paths must contain ''wave'' shapes at least on one of the diagonal nodes 22, 33 .. aa. The 3-wave has the "n" shape, the 5-wave looks like "NN", and in general the b-wave is 00~01..0b~1b~1a..10~20..2b..ba~bb. The shapes on the diagonal nodes will be rotated, but only in 2 out of 4 possible ways, depending on the parity of the row number.
	====Implementation====		====Implementation====
	The Perl program <code>[https://github.com/gfis/fasces/blob/master/data/gen_paths.pl gen_paths.pl]</code> uses the restrictions mentioned above.		The Perl program <code>[https://github.com/gfis/fasces/blob/master/data/gen_paths.pl gen_paths.pl]</code> uses the restrictions mentioned above.

imported>Gfis: path may not divide the square

2018-05-23T08:48:11Z

path may not divide the square

← Older revision		Revision as of 08:48, 23 May 2018
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	* We may always start with 00~01 without loss of generality.		* We may always start with 00~01 without loss of generality.
	There are 7 equivalent paths starting with 00~10, 0b~0a, 0b~1b, bb~ab, bb~ba, b0~1b, b0~a0. Therefore only one set needs to be computed.		There are 7 equivalent paths starting with 00~10, 0b~0a, 0b~1b, bb~ab, bb~ba, b0~1b, b0~a0. Therefore only one set needs to be computed.
	====Diagonal start and end====		====Diagonal start and end point====
	* Start and end point are in opposite corners.		* Start and end point are in opposite corners.
	With the normalized start 00~01, and 0b as end point, we get 00b~10b, and therefore 00~10, a contradiction. With b0 as end point, we get 0b0~1b0, and the same contradiction 00~10. Therefore, the end point must be bb.		With the normalized start 00~01, and 0b as end point, we get 00b~10b, and therefore 00~10, a contradiction. With b0 as end point, we get 0b0~1b0, and the same contradiction 00~10. Therefore, the end point must be bb.
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	* The path starts with 00, 01, 02, ...0b.		* The path starts with 00, 01, 02, ...0b.
	If there is any deviation 0n~1n with 0 < n < b, then the corresponding diagonal element 0nn has, in addition to its 2 normal neigbours in the adjacency matrix, a 3rd neighbour 1nn in the next z plane.		If there is any deviation 0n~1n with 0 < n < b, then the corresponding diagonal element 0nn has, in addition to its 2 normal neigbours in the adjacency matrix, a 3rd neighbour 1nn in the next z plane.
			====The path may not divide the square====
			* If the path hits a border, one of the 2 neighbouring nodes must already be contained in the path.
			If both neighbours were not visited so far, the path would divide the square in 2 halves, It would turn into one of them, and could never reach the non-visited nodes in the other one.
	====Implementation====		====Implementation====
	The Perl program <code>[https://github.com/gfis/fasces/blob/master/data/gen_paths.pl gen_paths.pl]</code> uses the restrictions mentioned above.		The Perl program <code>[https://github.com/gfis/fasces/blob/master/data/gen_paths.pl gen_paths.pl]</code> uses the restrictions mentioned above.

imported>Gfis: Straight left line

2018-05-22T20:09:14Z

Straight left line

← Older revision		Revision as of 20:09, 22 May 2018
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	A brute-force, backtracking algorithm which generates all pathes for a specific base runs rather fast up to base 5 (412 possible paths). For base 7, there are about 3 millions of paths starting with 00~01 ("~" denotes "adjacent to"), but only 47 of them can be continued in dimensions > 2. For base 9, the brute-force approach is not feasible, and it is necessary to develop restrictions for the backtracking algorithm.		A brute-force, backtracking algorithm which generates all pathes for a specific base runs rather fast up to base 5 (412 possible paths). For base 7, there are about 3 millions of paths starting with 00~01 ("~" denotes "adjacent to"), but only 47 of them can be continued in dimensions > 2. For base 9, the brute-force approach is not feasible, and it is necessary to develop restrictions for the backtracking algorithm.

	For ~~this~~ discussion, let digits b = base-1, a = b-1, c=b+1, h=a/2, and for elements (n, n) on the diagonal, m=n-1, p=n+1. For a node, we will write xy instead of (x,y), and zxy instead of (z,x,y). hh is the ''center'' element, {00, 0b, bb, b0} is the set of ''corner'' elements. Let st be the coordinates of the start point, and ef those of the end point.		For the following discussion, let digits b = base-1, a = b-1, c=b+1, h=a/2, and for elements (n, n) on the diagonal, m=n-1, p=n+1. For a node, we will write xy instead of (x,y), and zxy instead of (z,x,y). hh is the ''center'' element, {00, 0b, bb, b0} is the set of ''corner'' elements. Let st be the coordinates of the start point, and ef those of the end point.
	0b 1b .. ab bb .. .. .. .. ..		0b 1b .. ab bb .. .. .. .. ..

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	* Start and end point are in opposite corners.		* Start and end point are in opposite corners.
	With the normalized start 00~01, and 0b as end point, we get 00b~10b, and therefore 00~10, a contradiction. With b0 as end point, we get 0b0~1b0, and the same contradiction 00~10. Therefore, the end point must be bb.		With the normalized start 00~01, and 0b as end point, we get 00b~10b, and therefore 00~10, a contradiction. With b0 as end point, we get 0b0~1b0, and the same contradiction 00~10. Therefore, the end point must be bb.
	====~~Conflicts on the diagonal~~====		====Straight left edge====
	If 0n~1n~~, and if nn is not the endpoint~~, then ~~any of~~ the ~~4 possible neighbours of nn leads to a conflict:~~		* The path starts with 00, 01, 02, ...0b.
	~~''west'' mn~nn leads to conflict (0mn~~~0nn, ~~0nn~1nn)~~		If there is any deviation 0n~1n with 0 < n < b, then the corresponding diagonal element 0nn has, in addition to its 2 normal neigbours in the adjacency matrix, a 3rd neighbour 1nn in the next z plane.
	~~''south'' nm~nn leads~~ to ~~conflict (0nm~0nn~~, ~~0nn~~~1nn)		====Implementation====
	~~''east'' pn~nn leads to conflict (0pn~0nn, 0nn~1nn)~~		The Perl program <code>[https://github.com/gfis/fasces/blob/master/data/gen_paths.pl gen_paths.pl]</code> uses the restrictions mentioned above.
	~~''north'' np~nn leads to conflict (0np~0nn, 0nn~1nn)~~

	The Perl program <code>[https://github.com/gfis/fasces/blob/master/data/gen_paths.pl gen_paths.pl]</code> uses the ~~following considerations.~~



	~~and since , and for therefore always 0xy~1xy, which implies 0x~1x and 0y~1y. Assume that both the start and the end point are no corners. This implies 01~00~10, 0a~0b~1b, ab~bb~ba, and b1~b0~a0.~~

	~~A path is said to be ''continuing'' if it can be continued by the adjacency matrix rule into all (many) dimensions~~.

imported>Gfis: diag

2018-05-22T19:09:50Z

diag

imported>Gfis: Start/end opposite corners

2018-05-22T16:32:08Z

Start/end opposite corners

← Older revision		Revision as of 16:32, 22 May 2018
Line 87:		Line 87:
	A brute-force, backtracking algorithm which generates all pathes for a specific base runs rather fast up to base 5 (412 possible paths). For base 7, there are about 3 millions of paths starting with 00~01 ("~" denotes "adjacent to"), but only 47 of them can be continued in dimensions > 2. For base 9, the brute-force approach is not feasible, and it is necessary to develop restrictions for the backtracking algorithm.		A brute-force, backtracking algorithm which generates all pathes for a specific base runs rather fast up to base 5 (412 possible paths). For base 7, there are about 3 millions of paths starting with 00~01 ("~" denotes "adjacent to"), but only 47 of them can be continued in dimensions > 2. For base 9, the brute-force approach is not feasible, and it is necessary to develop restrictions for the backtracking algorithm.

	For this discussion, let digits b = base-1, a = b-1, c=b+1, h=a/2, and for elements nn on the diagonal, m=n-1, p=n+1. hh is the ''center'' element, {00, 0b, bb, b0} is the set of ''corner'' elements. ~~A path is said to~~ be ~~''continuing'' if it can be continued by~~ the ~~adjacency matrix rule into all (many) dimensions~~.		For this discussion, let digits b = base-1, a = b-1, c=b+1, h=a/2, and for elements (n, n) on the diagonal, m=n-1, p=n+1. For a node, we will write xy instead of (x,y), and zxy instead of (z,x,y). hh is the ''center'' element, {00, 0b, bb, b0} is the set of ''corner'' elements. Let st be the coordinates of the start point, and ef those of the end point.
	0b 1b .. ab bb .. .. .. .. ..		0b 1b .. ab bb .. .. .. .. ..

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	00 10 .. a0 b0 .. .. .. .. ..		00 10 .. a0 b0 .. .. .. .. ..
	~~It is natural to assume a start point 00~~, and ~~experiments show that~~ the ~~end point then always is the opposite corner bb~~ for ~~continuing paths~~. ~~But there are continuing paths which are not rotational symmetric~~ (~~e.g. Ls and Ln~~ for ~~base 5~~).
	====~~Where are the start~~ and end points?====		The path must visit every node in the square of the adjacency matrix exactly once, and so it must in the d-dimensional cubes (for d >= 3) of edge length ''base''. Furthermore, any projection of such a cube to the plane must show the same adjacency matrix (for odd bases; even bases are discussed later). In essence, any projection must not show some "T" pattern:
	The start point starts the multidimensional path, and the end point is the last path element in the square, 2-dimensional adjacency matrix. ~~The~~ end point continues into the 3rd dimension, ~~therefore~~ always ~~0xy~1xy~~, which ~~implies 0x~1x~~ and ~~0y~1y~~. ~~Assume that both the~~ start and the end point are ~~no corners~~. ~~This implies 01~00~10~~, ~~0a~0b~1b, ab~bb~ba~~, and ~~b1~b0~a0~~.		mn==nn==pn mp
			\|\| \|\|
			nm or mn==nn
			\|\|
			mm
			====Start and end points in some corner====
			The start point starts the multidimensional path, and the end point is the last path element in the square, 2-dimensional adjacency matrix. At the end point, the path continues into the 3rd dimension (z=1), It is natural to assume a start point 00, and experiments show that the end point then always is the opposite corner bb for continuing paths. BTW, there are continuing paths which are not rotational symmetric (e.g. Ls and Ln for base 5).
			* The start and the end point both are corner elements.
			Otherwise, they will show a "T" pattern in one of the projections zx, zy.
			* Start and end point are in opposite corners.

	====Normalization 00~01====		====Normalization 00~01====
	~~For all paths starting~~ with 00~01 ~~there is an~~ equivalent ~~path~~ starting with 00~10. Therefore only one set is computed.		* We may always start with 00~01.
			There are 7 equivalent paths starting with 00~10, 0b~0a, 0b~1b, bb~ab, bb~ba, b0~1b, b0~a0. Therefore only one set needs to be computed.
	====Conflicts on the diagonal====		====Conflicts on the diagonal====
	If 0n~1n, and if nn is not the endpoint ~~of the 2d path~~, then any of the 4 possible neighbours of nn leads to a conflict:		If 0n~1n, and if nn is not the endpoint, then any of the 4 possible neighbours of nn leads to a conflict:
	* ''west''~~ ~~ mn~nn leads to conflict (0mn~0nn, 0nn~1nn)		* ''west'' mn~nn leads to conflict (0mn~0nn, 0nn~1nn)
	* ''south'' nm~nn leads to conflict (0nm~0nn, 0nn~1nn)		* ''south'' nm~nn leads to conflict (0nm~0nn, 0nn~1nn)
	* ''east''~~ ~~ pn~nn leads to conflict (0pn~0nn, 0nn~1nn)		* ''east'' pn~nn leads to conflict (0pn~0nn, 0nn~1nn)
	* ''north'' np~nn leads to conflict (0np~0nn, 0nn~1nn)		* ''north'' np~nn leads to conflict (0np~0nn, 0nn~1nn)

	The Perl program <code>[https://github.com/gfis/fasces/blob/master/data/gen_paths.pl gen_paths.pl]</code> uses the following considerations.		The Perl program <code>[https://github.com/gfis/fasces/blob/master/data/gen_paths.pl gen_paths.pl]</code> uses the following considerations.



			and since , and for therefore always 0xy~1xy, which implies 0x~1x and 0y~1y. Assume that both the start and the end point are no corners. This implies 01~00~10, 0a~0b~1b, ab~bb~ba, and b1~b0~a0.

			A path is said to be ''continuing'' if it can be continued by the adjacency matrix rule into all (many) dimensions.

imported>Gfis at 10:26, 22 May 2018

2018-05-22T10:26:33Z

← Older revision		Revision as of 10:26, 22 May 2018
Line 87:		Line 87:
	A brute-force, backtracking algorithm which generates all pathes for a specific base runs rather fast up to base 5 (412 possible paths). For base 7, there are about 3 millions of paths starting with 00~01 ("~" denotes "adjacent to"), but only 47 of them can be continued in dimensions > 2. For base 9, the brute-force approach is not feasible, and it is necessary to develop restrictions for the backtracking algorithm.		A brute-force, backtracking algorithm which generates all pathes for a specific base runs rather fast up to base 5 (412 possible paths). For base 7, there are about 3 millions of paths starting with 00~01 ("~" denotes "adjacent to"), but only 47 of them can be continued in dimensions > 2. For base 9, the brute-force approach is not feasible, and it is necessary to develop restrictions for the backtracking algorithm.

	For this discussion, let digits b = base, a = b-1, c=b+1, h=a/2, ~~1 <= n <= b~~, m=n-1, p=n+1. hh is the ''center'' element, {00, 0a, aa, a0} is the set of ''corner'' elements. A path is said to be ''continuing'' if it can be continued by the adjacency matrix rule into all (many) dimensions.		For this discussion, let digits b = base-1, a = b-1, c=b+1, h=a/2, and for elements nn on the diagonal, m=n-1, p=n+1. hh is the ''center'' element, {00, 0b, bb, b0} is the set of ''corner'' elements. A path is said to be ''continuing'' if it can be continued by the adjacency matrix rule into all (many) dimensions.
			0b 1b .. ab bb .. .. .. .. ..
	It is natural to assume a start point 00, and experiments show that the end point then always is the opposite corner aa for continuing paths.
			0a 1a .. aa ba .. mp np pp ..

			.. .. hh .. .. .. mn nn pn ..

			01 11 .. a1 b1 .. mm nm pm ..

			00 10 .. a0 b0 .. .. .. .. ..
			It is natural to assume a start point 00, and experiments show that the end point then always is the opposite corner bb for continuing paths. But there are continuing paths which are not rotational symmetric (e.g. Ls and Ln for base 5).
	====Where are the start and end points?====		====Where are the start and end points?====
			The start point starts the multidimensional path, and the end point is the last path element in the square, 2-dimensional adjacency matrix. The end point continues into the 3rd dimension, therefore always 0xy~1xy, which implies 0x~1x and 0y~1y. Assume that both the start and the end point are no corners. This implies 01~00~10, 0a~0b~1b, ab~bb~ba, and b1~b0~a0.
	====Normalization 00~01====		====Normalization 00~01====
	For all paths starting with 00~01 there is an equivalent path starting with 00~10. Therefore only one set is computed.		For all paths starting with 00~01 there is an equivalent path starting with 00~10. Therefore only one set is computed.

← Older revision		Revision as of 19:09, 22 May 2018
Line 108:		Line 108:
	* The start and the end point both are corner elements.		* The start and the end point both are corner elements.
	Otherwise, they will show a "T" pattern in one of the projections zx, zy.		Otherwise, they will show a "T" pattern in one of the projections zx, zy.
	* Start and end point are in opposite corners.

	====Normalization 00~01====		====Normalization 00~01====
	* We may always start with 00~01.		* We may always start with 00~01 without loss of generality.
	There are 7 equivalent paths starting with 00~10, 0b~0a, 0b~1b, bb~ab, bb~ba, b0~1b, b0~a0. Therefore only one set needs to be computed.		There are 7 equivalent paths starting with 00~10, 0b~0a, 0b~1b, bb~ab, bb~ba, b0~1b, b0~a0. Therefore only one set needs to be computed.
			====Diagonal start and end====
			* Start and end point are in opposite corners.
			With the normalized start 00~01, and 0b as end point, we get 00b~10b, and therefore 00~10, a contradiction. With b0 as end point, we get 0b0~1b0, and the same contradiction 00~10. Therefore, the end point must be bb.
	====Conflicts on the diagonal====		====Conflicts on the diagonal====
	If 0n~1n, and if nn is not the endpoint, then any of the 4 possible neighbours of nn leads to a conflict:		If 0n~1n, and if nn is not the endpoint, then any of the 4 possible neighbours of nn leads to a conflict:
	* ''west'' mn~nn leads to conflict (0mn~0nn, 0nn~1nn)		''west'' mn~nn leads to conflict (0mn~0nn, 0nn~1nn)
	* ''south'' nm~nn leads to conflict (0nm~0nn, 0nn~1nn)		''south'' nm~nn leads to conflict (0nm~0nn, 0nn~1nn)
	* ''east'' pn~nn leads to conflict (0pn~0nn, 0nn~1nn)		''east'' pn~nn leads to conflict (0pn~0nn, 0nn~1nn)
	* ''north'' np~nn leads to conflict (0np~0nn, 0nn~1nn)		''north'' np~nn leads to conflict (0np~0nn, 0nn~1nn)

	The Perl program <code>[https://github.com/gfis/fasces/blob/master/data/gen_paths.pl gen_paths.pl]</code> uses the following considerations.		The Perl program <code>[https://github.com/gfis/fasces/blob/master/data/gen_paths.pl gen_paths.pl]</code> uses the following considerations.