https://en.wikipedia.org/wiki/Tautochrone_curve

Half-pipe

http://www.iitgn.ac.in/mcm/cd/mcm-2011/ ... /11199.pdf

Statistics: Posted by Cuchulainn — Yesterday, 7:54 am

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Statistics: Posted by Alan — Yesterday, 4:15 am

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* "proportionally" in an O(mgh) = O(mv^2) sense and reflecting the relative effects of linear and rotational inertia of the rolling ball.

Statistics: Posted by Traden4Alpha — Yesterday, 12:12 am

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"The Kingdom you see is Carthage, the Tyrians, the town of Agenor;

But the country around is Libya, no folk to meet in war.

Dido, who left the city of Tyre to escape her brother,

Rules here--a long and labyrinthine tale of wrong

Is hers, but I will touch on its salient points in order....Dido, in great disquiet, organised her friends for escape.

They met together, all those who harshly hated the tyrant

Or keenly feared him: they seized some ships which chanced to be ready...

They came to this spot, where to-day you can behold the mighty

Battlements and the rising citadel of New Carthage,

And purchased a site, which was named 'Bull's Hide' after the bargain

Statistics: Posted by Cuchulainn — July 23rd, 2017, 10:44 pm

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Mrs. Cuchulainn got the first one right (without any maths!)

Statistics: Posted by Cuchulainn — July 23rd, 2017, 10:32 pm

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And this one: why doesn't the bottom fall?

A family that plays together, stays together.

My guess is it does not have enough kinetic energy. You can put the motion in Lagrange's equation and solve.

Statistics: Posted by Cuchulainn — July 23rd, 2017, 4:35 pm

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Yes, the question is: why is the wavy one faster then the flat one?

The wavy one is not always faster everywhere; but the total transit time is less. If you know y' then you can compute the integral by finite difference etc.

If want to get from start to finish ASAP, then minimising the integral will be a

Statistics: Posted by Cuchulainn — July 23rd, 2017, 4:22 pm

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Statistics: Posted by outrun — July 23rd, 2017, 4:13 pm

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The question is? shortest travel time?

Statistics: Posted by Cuchulainn — July 23rd, 2017, 3:43 pm

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Statistics: Posted by outrun — July 23rd, 2017, 1:51 pm

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The Cross of Kepler contains a great secrets about the Universe, why and how?

What are the exact proportions of the Cross, how is it linked to a Magic Number that no one seems to understand in modern times?

Tips 1: Music of the Spheres!

What is the form of the curve in space-time in travelling from the top of the cross to the right side of the cross that is the shortest transit time?

It's not dissimilar to St. Brigid's cross (or Queen Dido's problem)

- One of the more commonly told stories is of Brigid asking the King of Leinster for land. She told the king that the place where she stood was the perfect spot for a convent. It was beside a forest where they could collect firewood and berries. There was also a lake nearby that would provide water and the land was fertile. The king laughed at her and refused to give her any land. Brigid prayed to God and asked him to soften the king's heart. Then she smiled at the king and said "
**will you give me as much land as my cloak will cover?**" The king thought that she was joking and, hoping to get rid of her, he agreed. She told four of her sisters to take up the cloak, but instead of laying it flat on the turf, each sister, with face turned to a different point of the compass, began to run swiftly, the cloth growing in all directions. The cloak began to cover many acres of land. "Oh, Brigid!" said the frighted king, "what are you about?" "I am, or rather my cloak is about covering your whole province to punish you for your stinginess to the poor." "Call your maidens back. I will give you a decent plot of ground." The saint was persuaded, and if the king held his purse-strings tight in future, she had only to allude to her cloak to bring him to reason. Soon afterwards, the king became a Christian, began to help the poor and commissioned the building of the convent. Legend has it, the convent was known for making jam from the local blueberries which was sought for all over Ireland. There is a new tradition beginning among followers of St Brigid to eat jam on 1 February in honour of this miracle.^{[20]}^{[21]}

Statistics: Posted by Cuchulainn — July 1st, 2017, 8:09 pm

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https://en.wikipedia.org/wiki/Banach–Tarski_paradox

and thus conceptually pack twice as many spheres into the same space with a bit of hypothetical cutting and reshuffling.

"The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points."

Maybe useful, maybe not.

Statistics: Posted by trackstar — June 23rd, 2017, 1:28 pm

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