By Vladimir Brezina
wherein the sum of an infinite number of alternating positive and negative terms, progressively decreasing in absolute magnitude, is the unity. (Borges would be pleased.)
A contribution to this week’s Photo Challenge, Converge.
By Vladimir Brezina
wherein the sum of an infinite number of alternating positive and negative terms, progressively decreasing in absolute magnitude, is the unity. (Borges would be pleased.)
A contribution to this week’s Photo Challenge, Converge.
Vladimir Brezina
... has kayaked the waters around New York for over a decade in his red Feathercraft folding kayak. He comes originally from (the former) Czechoslovakia and has lived in the U.K. and California before settling down in New York. He is a neuroscientist at the Mount Sinai School of Medicine in New York City.
Johna Till Johnson
... is a kayaker and technology researcher at Nemertes Research. She's an erstwhile engineer, particle physicist, and science fiction writer. She was born in California and has lived in Italy, Norway, Hawaii, and a few other places. She currently resides in New York City.
Johna Till Johnson on Freepaddling in the Ten Thousa… | |
Larry Jensen on Freepaddling in the Ten Thousa… | |
MomwithanMBTAmission on Pipes at Grand Central St… | |
MomwithanMBTAmission on Subways | |
Johna Till Johnson on Subways | |
MomwithanMBTAmission on Subways | |
TheresaBarker on Freepaddling in the Ten Thousa… | |
Johna Till Johnson on Freepaddling in the Ten Thousa… | |
TheresaBarker on Freepaddling in the Ten Thousa… | |
Johna Till Johnson on Freepaddling in the Ten Thousa… | |
bluebrightly on Freepaddling in the Ten Thousa… | |
Diana Szatkowski on Freepaddling in the Ten Thousa… | |
Johna Till Johnson on Freepaddling in the Ten Thousa… | |
Johna Till Johnson on Freepaddling in the Ten Thousa… | |
Anna Mallin on Freepaddling in the Ten Thousa… |
Fantastic!!
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Thank you!!
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It is so creative to use math to show the concept of converge. I have been interested in math all my life, so I took a few minutes to verify the accuracy of the formula in your post.
For alternative additions and subtractions, the left side of the formula can be simplified as (3/4 + 3/16 + 3/64 + …….). Then move entire left side over to the right side of the formula, which becomes 1 – (3/4 + 3/16 + 3/64 + …….). Step by step, it will become (1/4 – 3/16 – 3/64 – …..), then (1/16 – 3/64 – …..), then (1/64 – …..). Following the same sequence, it will become a 1/(multiples of 4), i.e., an inverse of an infinite number, which should be zero.
This gives the verification of the equation.
Am I right?
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Hi, Yan,
Sorry for the delayed reply! Here’s my take on it (it’s a JPG image: click on it to see a larger version)—
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I enjoy number theory. My wife thinks I am strange that way. I am going to assume that is a win-win scenario.
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It is definitely an acquired taste ;-)
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Love this
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:-) Thank you!
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Probably 75% of all the pictures I take are towards a vanishing point, what is being called convergence here. Yours are lovely :-)
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I don’t have too many pictures like that, surprisingly. So I was forced to use these :-)
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Ooh maths . . . my head hurts
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No need for that! Enjoy the photos :-)
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Pingback: weekly photo challenge: converge | Musings of a Random Mind
Great post & def, Vlad. Happy Holidays to you both!
del
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And to you, belatedly :-)
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It’s never too late!
:-)
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Wonderful!
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Glad you like it—thank you!!
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Methinks I saw one of these before? Love shots like this!
janet
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Yes, you saw it before, probably several times. Some photos are just too handy for multiple photo challenges :-)
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love the mirrors in mirrors converging … I would have expected converging waterways from you…
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I don’t have too many shots of waterways converging…. unfortunately, from kayak level, it’s hard to see the convergence :-)
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That is amazing! I love it. :-)
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Thank you, Eliz! :-)
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Excellent interpretation of this week’s challenge. You took me right back to my school days with the math…which I seem to have forgotten how to work out with formulas :)
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You may not have studied infinite series. They open up a can of worms, and so are usually studies fairly late…
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Actually, now that you say it, I might have come across infinite series during my university days. I think when you talk about infinite series, you study things like limits, geometric series, absolute convergence… I’m more of a creative kind of person, shying away from numbers… :)
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That sounds about right.
Of course, mathematicians would say—and I agree—that mathematics is very creative, since it creates a whole abstract world ;-)
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A clever interpretation Vlad :-)
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Thank you, Madhu! :-)
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amazing! why do these pictures make a curve in the distance?
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That’s a very good question, and the answer is not immediately obvious. But some back-of-the-envelope ray tracing suggests that it’s probably because the two mutually reflecting mirrors are not set into their respective walls exactly parallel to each other, or are themselves a bit curved.
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i see. very interesting!
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Brilliant! :)
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Thank you so much!! :-)
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That’s so cool. Maybe if math had been shown to us like this in school it would have been more fun and understandable for us math-challenged folks. :)
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Unfortunately, the formula is not equivalent to the photos in any precise way, so the photos don’t provide an illustration of the mathematics…
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Great idea Vladi !!
Great shots !
Ciao ciao
Max
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Thanks so much, Max!!
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Such an interesting photo, alongwith a mathematical formula….hahahaha….makes it so splendid…loved it..:)
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Glad you enjoyed it—thanks very much! :-)
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I’m sorry to say that you lost me on the math but I enjoyed the great pictures of you and Johna and you and Johna and you and Johna…..ad infinitum.
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Fortunately, we soon disappear round the bend, otherwise you might have too much of us :-)
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mirrors always surprise me!
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Especially if there is more than one…. :-)
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