Mathematically Correct Breakfast -- Mobius Sliced Linked Bagel

don't say http://facebook.com/theBagels never posts anything about Bagels...

It is not hard to cut a bagel into two equal halves which are linked like two links of a chain.

Mathematically Correct Breakfast

via georgehart.com

How to Slice a Bagel into Two Linked Halves

To start, you must visualize four key points. Center the bagel at the origin, circling the Z axis.

A is the highest point above the +X axis. B is where the +Y axis enters the bagel.

C is the lowest point below the -X axis. D is where the -Y axis exits the bagel.

These sharpie markings on the bagel are just to help visualize the geometry

and the points. You don't need to actually write on the bagel to cut it properly.

The line ABCDA, which goes smoothly through all four key points, is the cut line.

As it goes 360 degrees around the Z axis, it also goes 360 degrees around the bagel.

The red line is like the black line but is rotated 180 degrees (around Z or through the hole).

An ideal knife could enter on the black line and come out exactly opposite, on the red line.

But in practice, it is easier to cut in halfway on both the black line and the red line.

The cutting surface is a two-twist Mobius strip; it has two sides, one for each half.

After being cut, the two halves can be moved but are still linked together, each passing through

the hole of the other. (So when you buy your bagels, pick ones with the biggest holes.)

If you visualize the key points and a smooth curve connecting them, you do

not need to draw on the bagel. Here the two parts are pulled slightly apart.

If your cut is neat, the two halves are congruent. They are of the same handedness.

(You can make both be the opposite handedness if you follow these instructions in a mirror.)

You can toast them in a toaster oven while linked together, but move them around every

minute or so, otherwise some parts will cook much more than others, as shown in this half.

It is much more fun to put cream cheese on these bagels than on an ordinary bagel. In additional to

the intellectual stimulation, you get more cream cheese, because there is slightly more surface area.

Topology problem: Modify the cut so the cutting surface is a one-twist Mobius strip. (You can still get cream cheese into the cut, but it doesn't separate into two parts.)

Calculus problem: What is the ratio of the surface area of this linked cut to the surface area of the usual planar bagel slice?

For future research: How to make Mobius lox...

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Note: I have had my students do this activity in my Computers and Sculpture class. It is very successful if the students work in pairs, with two bagels per team. For the first bagel, I have them draw the indicated lines with a "sharpie". Then they can do the second bagel without the lines. (We omit the schmear of cream cheese.) After doing this, one can better appreciate the stone carving of Keizo Ushio, who makes analogous cuts in granite to produce monumental sculpture.

One Man’s Discarded Ticket Can Be Another Man’s Salary - NYTimes.com

Picking (Up) Winners Without Placing a Bet

Earl Wilson/The New York Times

Jesus Leonardo is a stooper, picking up tickets that others have thrown away at OTB parlors.

Once upon a wager on a race run at Aqueduct Racetrack in Queens, Mr. Leonardo, who lives in Wanaque, N.J., became a stooper by accident.

In 1999, he walked into that same OTB parlor in Midtown and placed a bet. He watched the race, was sure he had lost and threw away his Pick 3 ticket.

“But just as I was leaving, I looked up at the screen and realized an inquiry had been made,” he said, referring to a review of the race to check for possible rules infractions. “All of a sudden, the results changed and I actually won $900.”

He began a frantic search for his ticket, picking up hundreds off the floor, and from ashtrays and garbage cans. He could not find it, however, and began pleading with the manager on duty.

“She said there was nothing she could do about it,” Mr. Leonardo said. “I was so upset, almost in tears. Finally, she said, ‘Look, if you want to take the garbage home with you and look for your ticket, go right ahead.’ ”

He did. Although he did not locate his $900 jackpot, he found two other winners in the trash, worth a combined $2,000.

For the past 10 years, Jesus Leonardo has been cleaning up at an OTB parlor in Midtown Manhattan, cashing in, by his own count, nearly half a million dollars’ worth of winning tickets from wagers on thoroughbred races across the country.

Discouraged bettors usually toss their tickets as soon as the race they have wagered on has ended, which may be a bit premature.

Earl Wilson/The New York Times

Jesus Leonardo with a credit voucher for $6 that he earned in his “work” at an OTB parlor in Midtown Manhattan.

During his glorious run, Mr. Leonardo, 57, has not placed a single bet.

“It is literally found money,” he said on a recent night from his private winner’s circle. He spends more than 10 hours a day there, feeding thousands of discarded betting slips through a ticket scanner in a never-ending search for someone else’s lost treasure.

“This has become my job, my life,” he said. “This is how I feed my family.”

Leonardo, who favors track suits and wears his graying hair and bushy beard in long ponytails, is what’s known in horse racing parlance as a stooper — a person who hangs around racetracks and betting parlors picking up tickets thrown away by others. Most tickets are losers, but enough are winners to make it worth his while.

To his stable of OTB buddies, Mr. Leonardo is the Secretariat of stoopers.

“He’s a legend,” said Paul Pepad, 57, an out-of-work musician who lives in Manhattan. “Everyone knows that this is his turf, that all the tickets thrown out belong to him, period. It’s just been that way as long as I can remember.”

T.D. Thornton, a journalist who wrote about stoopers in his 2007 book, “Not by a Long Shot: A Season at a Hard-Luck Horse Track,” said: “Stoopers are the gleaners of the racetrack world. Stoopers have a relationship with horse tracks that goes back to the advent of parimutuel betting in the early 1930s. There is an unwritten code in racing that says stoopers are tolerated as long as they are not perceived as harassing or stalking customers.”

“They are allowed to live on the fringes,” he added.

Mr. Leonardo, who is married with two teenagers, is hardly living on the fringes. He said that stooping brings him $100 to $300 a day, and more than $45,000 a year. Last month, he cashed in a winning ticket from bets made on races at Santa Anita Park in Arcadia, Calif., for $8,040. His largest purse came in 2006, when he received $9,500 from a Pick 4 wager (choosing the winners of four consecutive races) at Retama Park Race Track in Selma, Tex.

It is all taxable income. “I file my winnings with the I.R.S. every year,” Mr. Leonardo said in his thick Dominican accent.

Freddy Peguero, 53, a short-order cook from Manhattan, rooted for Mr. Leonardo to scan a winner one recent afternoon.

“Everybody in here loves Jesus,” he said. “When Jesus wins, we all eat, and we all drink. Jesus is a very generous man.”

Once upon a wager on a race run at Aqueduct Racetrack in Queens, Mr. Leonardo, who lives in Wanaque, N.J., became a stooper by accident.

via nytimes.com

Islamic Default

Interest-free ‘sukak’ bonds at first withstood the economic crisis, but Dubai’s Nakheel default has hurt their reputation
By Lorne Cutler

With $25-billion in Dubai debt potentially in default, much of it on the spectacular Palm Jumeirah – the palm-shaped island off this Middle East emirate --

the ramifications of Dubai’s debt problems go far beyond the potential for global contagion. A default will also test the true nature and performance of Islamic financing under stress. This critical area to watch has never really been tested before.

Islamic financing has been growing rapidly in importance in many Muslim
countries as well as in Western countries with large Muslim
populations. With almost a trillion dollars in assets worldwide,
Islamic financing has attracted the interest of Islamic banks and
conventional banks alike, particularly in Dubai.

Dubai
World (DW) is the holding company for the business assets of the
Government of Dubai. While most of the debt incurred by DW and its
subsidiaries was conventional, a significant amount was Shariah
compliant. Of DW’s $60-billion in debt, Nakheel, one of Dubai’s largest
real estate developers, issued $5.2-billion in Islamic bonds of which
$3.5-billion is due this month. Nakheel and DW are not able to pay this
debt and are trying to reschedule.

As interest is not
allowed under Islamic law, three broad financing structures have been
developed by Islamic scholars, all involving bank purchases of the
goods that the buyer wishes to procure. The bank will then either
resell the goods to the buyer (Murabaha), lease them to the buyer
(Ijarah), or go into partnership with the buyer with the bank putting
up the money and getting a return only if the venture is profitable
(Musharaka). While lenders have modified these structures to minimize
their exposure to the underlying commercial transaction, Islamic loans
must obtain the approval of Islamic scholars. It is often controversial
as to what structures are truly Shariah compliant.

The sukuk
or Islamic bond was further developed to facilitate Islamic financing.
This instrument allows the debt to be securitized and syndicated, while
the underlying financing must follow one of the acceptable Islamic
structures. Sukuks are traded on the Nasdaq Dubai exchange. Elaborate
sukuks have been developed to mitigate risk and provide returns similar
to that of conventional interest-bearing loans.

Existing
Nakheel assets were sold to a special purpose vehicle (SPV) and the
assets were then leased back to Nakheel. The SPV raised the funds to
buy the assets by issuing lease-based sukuks. The proceeds of the sukuk
were used to fund construction of the Palm and the sukuk holders were
to be paid out of Nakheel’s lease payments to the SPV.

The
major strength of sukuks was also their major weakness. There had been
no history of sukuk defaults. While this was very reassuring to lenders
it also meant that there was no experience as to how the Dubai courts
would treat a sukuk in default.

The potential default of
Nakheel’s sukuk raises key issues for both sukuk holders and proponents
of Islamic financing. While structured according to Shariah law, sukuks
are typically governed by U.K. law and international arbitration. There
is no experience as to how the courts of Dubai will view these bonds
and whether U.K. or Shariah law will take precedence. The prospectus
warned that the Dubai courts are not bound to enforce U.K. judgments
without re-examining the merits of the case and may not consider U.K.
law as the governing law. If Islamic lenders are supposed to risk
share with the borrower, the courts may consider Islamic loans lower
priority to Nakheel’s or DW’s conventional debt. Lenders would probably
rather see these loans restructured than be pursued through the
courts. Acceptance of DW’s waiver of sovereign immunity also needs to
be watched.

Different Islamic scholars have different
interpretations of what is permissible under Shariah law. Nakheel’s
sukuk was approved by the Dubai Islamic Bank’s Shariah board. It is not
known, however, as to how the Dubai courts would react should some
other Islamic scholars come forward and make a case that the loans were
never Shariah compliant. Nakheel’s sukuks were secured by existing real
estate developments and land. Even if this security is enforceable,
with the 50% decline in real estate values in Dubai, the security is
not worth what it once was. If the sukuk structure and security cannot
stand up in court, Dubai’s ability to attract conventional and Islamic
financing will be jeopardized.

The other challenges posed by
the current situation are to the religious scholars. Islamic scholars
tend to view Islamic financing as being morally superior to
conventional financing due to its prohibitions against interest and the
taking of undue risk. As last year’s banking crisis for the most part
bypassed Islamic banks, this sense of Islamic financing’s moral
superiority was reinforced for many.

The default of Nakheel’s
sukuk clearly shows that Islamic banks are not immune to taking undue
risk and conducting insufficient due diligence. While real estate
bubbles can form anywhere, Dubai’s was more obvious than most. Islamic
lenders were no different than conventional lenders in ignoring the
growing real estate bubble and will now have to re-evaluate their
perception of risk the same as conventional lenders. Such reflection
may result in a shift to musharaka financing structures which are often
considered to be more acceptable by Islamic scholars. The risk of
default is lessened as debt payments are only due if the venture is
profitable but the risk of loss is higher. If this reduces the use of
less risky murahabas and ijarahs, Islamic banks could lend less and be
less profitable.

Regardless of the outcome of the Dubai debt
crisis, this crisis will force changes in how both conventional and
Islamic banks consider risk in Shariah financing.

Financial Post
lacutler1@hotmail.com
Lorne Cutler is an Ottawa-based independent financial consultant.

Photo: The Burj Dubai, man’s tallest structure: There’s never been a default of interest-free debt. Which means there’s no precedent. (Getty Images)

via network.nationalpost.com

UN Resolution 1701