omenti

the context and quality of data

Passing white light through a prism to break it into a range of colored light, Newton introduced the word spectrum into optics.  The colored apparition was packed with information contained in the light - a mixture of frequencies and intensities.  Eventually it was learned that studying these revealed important aspects of matter the light had interacted with.  In the mid 1800s physicists reignited the sleepy field of astronomy by attached spectroscopes to telescopes allowing the study of what made the starlight.  Among the discoveries was an unknown yellow line in the Sun's spectrum.  It was proposed as a possible new element later called Helium in honor of the Sun. 25 years it was discovered on Earth.  

 

The technical is powerful and commonly used in many areas of science, medicine with thousands of commercial applications. The caveat is you need to use the right technique and appropriate instrument. 

 

The heart of many types of spectrometers is simple - a well characterized and calibrated light source, something to hold a sample, a prism or diffraction grating to break up the light into a spectrum and a way to measure the intensity of the spectrum as a function of its wavelength (color, sort of).  A smartphone has enough processing power to perform serious analysis and a few companies have made low cost spectrometers for teaching purposes.  A good example is this $400 unit from Pasco.  It lacks the precision to do serious work, but its great fun and works with an iPhone or iPad.

 

Three times in the past two years people have sent descriptions and pitches of even more portable units using integrated spectrometer modules.  While they might be fun, these usually do reflected light surface spectroscopy and appear to have poor resolution.  Silly claims are frequently made - that you can determine what is in your medicine, do food safety and calculate the calories in a meal.  

 

Nope - it isn't going to work like that.  You can't use out of context uncalibrated data and somehow perform cloud magic with 'algorithms' ... Success during a Kickstarter funding period does not guarantee impossible magic.  There will be a flood of interesting sensors and even serious instrumentation for smartphones and some of it will enable a density of information that is potentially transformative. Consider thousands of accurate pollution monitors per square kilometer in cities.  China is ripe for a revolt from a data informed citizenry.

 

Data, even the simplest data, has context.  What was the sample. how were the measurements made, what is the accuracy of the measuring tool, did the accuracy change over time, how was the data preprocessed, how was it postprocessed...?  

 

A few years ago I came across a detailed description of a rope transmission from Bordeaux, France that moved mechanical power from a water wheel to several businesses hundreds of feet away.  These contraptions sometimes often had mechanical efficiencies that still impress.  My answer seemed far too low.  I smugly came to the conclusion that the book must be in error.

 

The problem resurfaced recently when I was reading about the spread of the metric system in France.  Standardization of weights and measures was of vital importance to commerce as hundreds of local systems were being used.  The author noted rural Bordeaux defined a foot using a standard that was 35.7cm long - dramatically different from the now standard definition of about 30. 5cm.  I had been doing my calculations based on a bad assumption.  Using this definition a six foot tall person would be a bit over five foot one...

 

About three years ago a friend came to me with a question about a distribution of measurements.  She's quite tall and was thinking about putting together a specialized shop for tall women.  An important early question was to understand the potential market size.  While there are many factors that go into clothing size, height is a reasonable proxy to get a rough idea.  She knew there were tables showing height distributions,  but it wasn't clear to her how to navigate the information.  I had been involved in a large anthropometry survey in the nineties so one thing led to another...

 

You can make a histogram of heights - 100 people at 64 inches, 125 at 65 inches and so on. Increasing the number of people in your sample will make the distribution look smoother.  With a very large population and very small measurement increments you'll get something like this:

 

It isn't exactly the bell-shaped curve you probably expected.  The problem is it is the composite of two two curves.  Men are about five inches taller than women on average and need to be considered a separate population.   Plotting men and women separately gives two bell curves. Add them and the original distribution appears.

 

 

This type of curve is a Normal distribution - anyone with a STEM background has studied them in depth as they turn up almost everywhere.  Human height turns out to approximately follow this type of distribution making it easy to calculate the distribution of potential customer heights assuming the published curve is relevant to your population.  Measurements have been made in most countries and  are repeated at regular intervals.¹ 

 

My friend's  target market was women who stand more than six feet tall.  A small market, but she and her friends were underserved so it made sense to investigate.  Perhaps the Internet would allow her to find enough customers to justify production runs greater than a few pieces.  She chose a huge American survey and found wrote a simple program that calculated how many women were predicated to be at or about a height 

from her email:

 

Something is broken Steve  I just integrate a normal distribution with a mean of 64.3 and a standard deviation of 2.5  from a height through infinity to get the fraction at and above a height.  Here are my numbers with a little rounding.  Any ideas?!?

 

72"    1 in 966

73"    1 in 3,900

74"     1 in 19,150

75"     1 in 107,000

76"     1 in 697,200

77"     1 in 5,298,900

78"     1 in 47,022,800

This is SO WRONG!  The number of girls in college basketball and volleyball are instant disproofs!

 

To produce a clean fit it standard practice to exclude the tails of a measured distribution and tell the reader where it is valid.  It wasn't published in this case,  at least not prominently, but the fit was only good to about 5'11.  The tails on the real curve tend to be thicker, but making a large enough measurement to smooth them out is unrealistic.  It is possible to get a good enough fit for all but a tiny fraction of the population with only 100,000 or so people.

 

Now I was curious.  Would it be possible to find a better fit to the distribution?  Over a period of a few months I found a few ways to build a more accurate distribution function.  Rather than a simple normal distribution it is the sum of three separate distributions.  The tricky part was calculating the errors.

 

I won't list all of the results, but I'm 95% confident that a 6'3" woman (my friend's height) is between 1 in 16,070 and 18,070 with 1 in 16,950 being the most likely.²  1 in 17,000 is probably a good enough starting point and better than any of the major surveys.  My tall friend Colleen would be about 1 in 267,000, but the bound is much larger between 1 in 219,000 and 1 in 420,000.  Another good friend is 6'8,  the estimates are very sketchy at her level:  1 in 1.38 million with a range from 1 in .76 million to 1 in 2.79 million.  Counting athletes as a lower bound is probably better:-)

 

The final plot looks identical to the second plot as the size of the change is only 0.02% of the regular normal curve.  This is a feature as the densely populated portion of the distribution function is nearly identical to the original. To get a sense of the shape of the adjusted function,  imagine a world where my corrections are 2,500 times larger making their contribution equal to that of the normal female height distribution.

 

                         

 

 

 

The new female probability curve is red and the old uncorrected female curve is blue in the first plot and the uncorrected male curve is blue in the second.³  In this different world a quarter of the female population is six feet tall or more.  One in one hundred and thirty would be at least Colleen's height.  This  alternate world could be fodder for social commentary fiction.  Tall women certainly wouldn't have problems finding clothing and female sports would probably dominate.  Perhaps height would be unimportant socially. 

 

Continuous distribution functions derived from observed data are common and, when used correctly, powerful.  It is all too easy to miss a fundamental assumption and come to the wrong result while feeling good about it.  Separate sanity checks, like looking at the roster of college volleyball teams, need to be normal practice.  Being playful and not be easily seduced is a requirement.   

 

The clothing concept turned out to be impractical.  Quality sources for reasonable costs simply don't exist for the necessary low volumes.  It became clear that made to measure is necessary. Some is done for men's shirts in Asia and South Asia, but costs are currently high.  In addition to height, there was too much variability in sizes to make the effort practical... at least for now.  The trick is mass made-to-measure.  Eventually it will be standard practice and fit will become nothing more than a memory. Until then some segments of the apparel market are tough sledding.  Her advice for anyone dealt an usual body: learn to sew, find a good tailor and make enough money for a few quality custom pieces. Find your own style rather than following the trends.

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¹ Health organizations usually take the measurements as they are a proxy for human health.  Cheating I copy and paste from an email I sent on the subject:

We tend to pay a lot of attention to numbers without worrying about their accuracy. Height is a good example. If you measure a large number of adult males and females, their heights will distribute themselves along a bell shaped curve. Most people will be close to an average (a mean). The larger the sample the more the distribution will take the shape of a bell shaped curve called the normal distribution. Remarkably all you need to know is the average and a number that tells you something about how rapidly the curve changes shape. From this a designer will know the height distribution of potential customers and can adjust their manufacturing plans accordingly. You can do similar exercises for many measurements - shoe size for example. 

The characteristic curves vary regionally. The US is very similar to England and can be taken as the same within the accuracy of most measurements. Northern Europe is taller, with Holland being the tallest. The North of Holland is particularly tall and is sometimes treated separately. Asia and South Asia are shorter than the US and South America is much shorter. Changes in mean and standard deviation are studied over time as height is a good proxy for health within a population. A tall person is not necessarily healthier than a short person, but for a child about 20% to 25% of the variation of their final height is influenced by health and nutrition rather than genetics.  If the health of a nation changes, so will it’s average height. Holland has seen a dramatic change in the 20th century - particularly after WWII with Northern Holland being the tallest region in the World.  For adults under 35 female height is only an inch less than mean male height in the US and mean male height approaches 6'2.

² A quick comment - the curves are probability distribution functions.  The horizontal axis is height, the the vertical axis is probability, so you get a sense of roughly what it is.  The exact probability for a point is meaningless.  The probability of someone being 73.0000001 is very nearly zero.  Just integrate the function over the range you are interested in. 

Measurement errors in the height studies and insufficient statistics are the major components that make up the errors.  A trained person using a simple stadiometer - those vertical rulers with a sliding horizontal arm - can repeatedly measure height to about a half centimeter, but some surveys are self-reported.

³ Here I have plotted real probabilities for each inch of height and have connected them with a smooth curve - a similar to the density function, but a bit different.  I'm just doing it to show there are different ways to interpret and display the data and drawing conclusions requires some understanding.

Here is the probability plot of just the correction in black scaled up to represent a full population next to a normal female plot in red.  The shape isn't simple and has a much larger mean and width than the normal curve for women's height.  Scaled down by a factor of 2,500 and added to the normal women's distribution it gives a "fat tail" that squares with other measurements.

 

 

Playing games like this make it easy to spot errors in your thinking or calculations. At some level this is just a game.  A good thing as you can spend a lot of time getting everything to work.

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Recipe Corner

 

This is modified from an old recipe from a cooking club I was part of

 

Indian Cauliflower with Several Spices

 

Ingredients

 

° 1 large head cauliflower, trimmed and separated into florets 

° 1 large yellow onion, thinly sliced

° 1 tbsp olive oil (you really should use coconut oil, but I didn't have any)

° 1 tsp mustard seeds 

° 1 tsp cumin seeds 

° 1 tsp nigella seeds (I didn't have so I skipped)

° 1 tsp fenugreek seeds 

° 1 tsp fennel seeds 

° 6 ounces tofu, stirred 'til smooth

° 1 tsp paprika

° ½ tsp cayenne

° ½ tsp turmeric

° juice of a half lemon

° kosher salt to taste

Technique

° heat the oil in a pan and add the seeds. When they dance and begin to darken, add the onions.

° sauté the onions until  translucent.

° add the cayenne, paprika and salt. 

° add the cauliflower florets and mix well.

° add the tofu, mix. Cover the pan and let the cauliflower cook until somewhat tender. Stir every few minutes

° add salt and lemon juice and mix .

 ° serve warm

 

summing things up

In high school you probably learned about the properties of infinite series.   Things like the sum of the reciprocals of the powers of 2: 1/1 + 1/2 + 1/4 + 1/8 + 1/16 + ^... converges on 2, but the sum of the reciprocals of the positive integers diverges: 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + ^... , growing to infinity.  There are any number of interesting convergent series related to really interesting numbers like e and Pi, not to mention it is useful to know when something blows up to infinity.  Math just works, eh?

 

 

But wait...  

 

A friend sent a link to a video that seemed wrong.  It made a completely nonsense claim: that the sum of all positive integers was not infinity.   1 + 2 + 3 + 4 + 5 + ... + infinity.   Stranger yet was the claim it was not an integer and also negative.   The video came from real mathematicians who sounded confident.  He looked for an April first posting date, but didn't find anything suspicious.   This had to be some kind of joke.  What was the trick?

 

 

 

It turns the video is correct.  In fact you need to be able to do sums like this if you want to use the Standard Model in physics as well as string theory and a few other areas.  The same folks put out a second video with a bit more math, but still at a high school level.  If you're really curious take a look.

 

Something extremely interesting is going on.

 

Imagine math in the days when the only numbers people used were real and rational.  Some clever person came along and started using square roots.  The square root of 1, 4, 9 and 16 made sense, but 2 was a problem.  You could approximate the square root of 2, but it wasn't a rational number.  A new type of math had been discovered that was not only was in another context, but expanded the notion of what math was.

 

On the surface the square root of a negative number seems pure nonsense - so much that the operation was defined as invalid on that side of the number line. But its just math, so why not?  The square root of a -1 was defined to be a new type of number - an imaginary number.   A new number system had been invented.  It was shown to be rigorous  in the proper context with certain definitions and has enormous value in physics and all of engineering.  

 

This is the story of math.  Rather than a pristine temple where people work with pure logic using only existing truth to find new truths, it is a bubbling sea of conjecture and relation that can lead to proof.  Ultimately a rigorous path needs to be found,  but the frontiers are messy.  Change comes with rebellion.

 

Leonhard Euler was just such a rebel.   He wondered if  sums that diverged to infinity had deeper meanings.   Some, like the sum of all positive integers, had a regular structure.  How might they differ from other infinite sums?  Was there a story different from the final tabulation at the end?

 

He came up with an ingenious technique and showed the sum of positive integers was -1/12 in a fashion not too far from that of the first video.  The first time I saw it in college was electrifying.   It was like sorting through an infinite pile of dirt and rock to find a tiny diamond.  When a similar series appears in quantum electrodynamics - an area of physics that gives us the highest precision of all measurements to date - you substitute in the little diamond and, low and beware, the calculation produces a meaningful result.  The most accurate measurements humans have made agree with a theory that makes use of this type of sum.

 

Euler worked out several other series.  The sum of the squares of all positive integers turns out to be 0 and that of the cubes is -1/120.  In fact the all of the sums of even powers of the the integers is 0 and the odd powers is not.  (Impress your friends by noting the sum of x_i¹⁰ over i = 1 to infinity is 0.)   He was unable to prove this rigorously.  That had to wait about a century for Bernhard Riemann who was able to show a deep connection with something that fascinates to this day - the Riemann zeta function.

 

So what seems both wrong and crazy turns out to make perfect sense in a certain context - a context that is fundamentally related to how we know the physical world at the deepest level.  Why does Nature seem to speak math?  Now there is a deep mystery... 

 

I should stress that the equivalence here is the core  issue.  Equivalence doesn't mean equal.  This gets deep into the analytic continuation of the Reimann Zeta function at s = -1 ... which happens to correspond to the sum of the positive integers.  The R-Z function needs to be extended through analytic continuation to be valid for -1.  The videos don't stress it enough.  Doing this you get the R-Z (-1) = -1/12 and also note that the sum of R-Z for -1 looks like the sum of positive integers.  Using this redefinition physics happens to work -- that is the magic.

 

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Recipe Corner

 

I don't know where the okra is coming from these days, but some is very high quality. I threw together a sort of crispy bhindi.  Not exactly traditional, but delicious.  As usual the quantities are recollections of what I think I did

 

Crispy Okra

 

Ingredients

 

° a pound of okra with the tops trimmed and sliced lengthwise into thin strips

° 1-1/2 tsp red chili powder

° 1-1/2 tsp garam masala

° 1/2 tsp turmeric

° 2 tbl chickpea flour

° 1 tsp cornstarch

° juice of one lemon

° vegetable oil

 

Technique

 

° mix the chili powerder, garm masals, turmeric, chickpea flour and cornstarch and toss the okra in it moistening with the lemon juice 

° heat some vegetable oil in a wok to something under its flash point and fry a handful of okra at a time until golden.  Drain on paper towels and let cool.

° I tried a second frying on some which makes them even crispier -- experiment!

 

 

digging in a bit deeper

 

42

 

It was Jean's birthday last week.  When I know someone's age I sometimes send a few examples of where the number comes up.  There was the beautiful Orion nebula M42, a musical based on a street in New York, the atomic number of molybdenum, the critical angle that describes where a rainbow appears and, of course the answer to the ultimate question of life, the Universe and everything.¹  Jean liked the rainbow as anyone should, but today it occurred to me that 42 was a term in an interesting sequence.

 

In number theory there is a set of numbers where the sum of the inverses of the prime divisors plus the inverse of the number is equal to 1.  Not many numbers have this property, but a few do

 

    2  the prime factor is 2:  1/2 + 1/2 = 1

    6 the prime factors are 2 and 3:  1/2 +1/3 +1/6 = 1

    42 the prime factors are 2, 3, and 7:  1/2 + 1/3 + 1/7 + 1/42 = 1

 

Jean's year...

 

The next few terms are 1806 and 47058 which you can verify for yourself, but after that the numbers grow rapidly.  

 

Neil Sloane is a friend from the Bell Labs days who loves integer sequences - as a passion he built and maintains the world's archive of them - OEIS.  I typed in a few terms and there was my sequence - primary pseudoperfect numbers.  So Jean - your age is the third primary pseudoperfect number.  Possibly a catchy tshirt?

 

There are an infinite number of integer sequences - some are incredibly useful and others incredibly obscure.  Math doesn't have a requirement that sometime be useful.  It is nice if it is beautiful and deep, but application to the world is for others to worry about.

 

If you hear the phrase mathematical sequence you probably think about the Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and so on.  The sequence is defined by F_n = F_n-1 + F_n-2 where F₁ = 1 and F₂ =1.²  There are any number of beautiful mathematical properties associated with the sequence and it is often used to describe some constructions in Nature - the patterns of seeds in a sunflower, the spiral of waves,  even the breeding of rabbits.  My eighth grade math teacher loved the sequence and pointed out that the ratio of adjacent terms gets closer to the golden ratio as the series progresses.  For him the golden ratio (φ) was deeply linked to beauty in nature and art.  I remember writing out the first twenty or so terms and taking the ratio watching it slowly move towards 1.6180349...  (it didn't get that close in the first twenty terms).

 

For a several years I thought the Fibonacci sequence defined φ.  Then I saw something and could ask a deeper question.  Here's a start.  Pick a couple of random numbers - for simplicity make them positive integers. Add them and then proceed using the rule X_n = X_n-1 + X_n-2 .. related to the Fibonacci sequence, but you are providing your own seed with the random numbers.  So if I picked 1435 and 3956 the third term would be 5391, the forth would be 9347 and so on.  Do this at least a dozen times and then take the ratio of the last term and that immediately preceding it.  Cute, eh?  If you program you may want to make a simple program that shows the convergence.

 

Now I want to do a bit of math, but the tools to write equations that will render on the range of browsers are awful - so I'll get out a pen and sheet of paper.  Nothing sophisticated, so don't worry.

 

 

Any two integers used as a seed will produce a sequence that converges to the golden ratio.  The Fibonacci sequence is an interesting subset of an infinite number of such sequences.  Math rules. Just play and questions can lead to insight.  Some of these are seen in Nature, but most of the Fibonacci relations are fairly weak.  Oh well...

 

For fun consider Friedman numbers ...  numbers that can be written as a mathematical expression using the digits of the number and simple arithemetical opertions like brackets, +, -, *, / and ^. 

 

25 = 5²

121 = 11²

125 = 5⁽²⁺¹⁾

125 = 21*5

 

many many more.  Some preserve the order of the digits and are called nice Friedman nub:

 

127 = -1 +2⁷

6455 = (6⁴ -5)*5

 

repeating digits are fun

 

99999999 = (9 + 9/9)^{9 - 9/9} - 9/9

11111111111 = ((11 - 1)¹¹ - 1*1)/(11 - 1 - 1)

 

Other than simple 4 digit and smaller numbers I have found very fewon my own, but they have a certain cool beauty.  Check this one out - not only does it have one of all of the digits from 1 to 9, but it is nice.

 

268435179 = -268 + 4^{(3*5 -1^7)} - 9  (note my text editor doesn't allow double superscripting, so I use a power sign)

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¹ If you haven't read or listened to Douglas Adams' wonderful The Hitchhiker's Guide to the Galaxy just drop whatever you're doing, get a copy and enjoy.  My favorite version is the BBC Radiophonic play studio recording - your library may have it or you can check the normal places - ahem.  I can't listen to 'The Journey of the Sorcerer' from the Eagle's One of these Nights album without thinking about it.

The quick summary of the number is a group of hyper-intelligent beings build Deep Thought, a special computer designed to come up with the answer to the ultimate question of life, the Universe, and everything... Deep Thought cranks away for about 7.5 million years and gives the answer ---  42.  Of course the hyper-intelligent beings didn't know the question.  Getting at that required building the Earth...

² I'm keeping the convention you probably learned.  In modern usage the sequence starts with 0, so F₀ = 0 and F₁ = 1.

 

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Recipe Corner

 

I had to have some corn soup.  This is a modification of an old recipe I've been using - you can literally make it a heart stopping dish with heavy cream, but the recent version omitted it and was just fine.  It would be better with fresh sweet corn, but this is December and I used a good canned variety.  You might want to think about a garnish if you are serving this.  Roasted corn with cayenne, basil, ...  any number of garnishes would work.

 

Corn Soup

 

Ingredients

 

° olive oil (about 2 tbl)

° 1 bunch leeks, sliced thin

° 2 carrots, chopped

° 3 cups of corn

° 4 cups vegetable broth (homemade is best - unsalted is always the way to go)

° 1/2 cup white wine

° 1/2 tsp cayenne papper

° 1 tsp cumin

° 1 bay leaf

° kosher salt

° freshly ground black pepper

° 3/4 cup heavy cream if you dare

 

Technique

 

° heat a glug (maybe 2 tbl) olive oil over medium heat.  Add the leeks and carrots and sauté until tender.  Add corn and continue to cook until tender/

° Add the wine and bring to a simmer.  Cover and cook until the liquid is almost completely reduced (8 minutes for me) Add the broth and bring back to a simmer.

° Season with the bay leaf, cumin, cayenne, salt and pepper.  Simmer for at least 30 minutes to allow a flavor to develop.

° Remove the bay leaf and purée in a blender or (better) with an immersion blender.  

° Check the seasoning.  If going for a rich soup add the heavy cream here and return to simmer.

° Serve and possibly garnish

 

elements of style

Thomas Jefferson had the reputation of an extreme bibliophile, but was forced to make choices 'only' reading about 7,000 books.    Ben Franklin had a similar information challenge and both men were unable to fully respond to their mail.  They lived at a time when it seemed like a powerful and well connected mind could have a deep understanding of general knowledge, but that was an illusion.  Information growth had become too great for single humans to manage and it had become necessary  to construct filters and trust the filters of others.

 

Today we have something of an information explosion.  Physics occupies a tiny percentage of humanity, but is sometimes citied as an extreme example of information processing.  Single particle physics experiments generate information at over 100,000 times the rate it can be stored.  A carefully designed hierarchy of filters known as triggers identify relevant information and throw away the rest.  

 

Although not as dramatic as a particle physics trigger, most organizations have developed a series of filters - some are carefully considered while others are poorly understood.  We might think of our social and work interactions to be well understood, but in reality they are poorly understood making it difficult to model.¹

 

Our senses are even more overloaded - we live at the bottom of an information well that our brain partially samples and heavily filters to create an illusion, albeit a rich one, of reality.  We tend to trust these personal filters, but careful science shows we are easily deluded and caution is necessary.

 

Filters are central to our relationship with the natural, cultural and machine worlds.  Companies like Google, FaceBook, and many others build filters that may not be aligned with the user interests and it is usually impossible to understand what these filters are.  Some focus on adding value to advertising and attempt to drive consumption and perceptions about our place in culture.  The difference between fashion and style presents an interesting example.  

 

Fashion is the relationship to the external.  It is how you mix and match predefined pieces to show you are part of a larger tribe that is constantly in motion.  It can be expensive or cheap, but is ephemeral.  It allows you to distance yourself from yourself and blend into a cultural tribe.  It is 100,000 people wearing the same jeans, shoes and a tshirt that proclaims 'I am an individual'.  

Style is the relationship to the internal. It is about your sense of self, your identity, your perception of who you are. It is a process of introspection where you ask what is in here and reflect that in what you wear.  The emphasis is on the person rathe rather than popular culture and social norms.  It requires a good sense of self.

 

Jheri and I have an ongoing conversation on the difference between style and fashion.²  With her permission I quoted her definitions from a very long letter describing her continuing journey finding her personal style.  Her body type is unusual enough that nothing fits off the rack.  She described her experimentation with fashion during her teenage years as learning about 'hats, scarves and cheap costume jewelry' as well as what looked good on her friend.  By the time I met her she was having clothing modified, but she was also searching out pieces that expressed her in one way or another.  Pieces that would last a long time.  Her closet is not deep, but her look can be striking and reflects her personality and image of herself.  The image is some out of season boots that were slightly damaged.  The calf width was too large, but she saw something and figured out an usual way to wear them.  She also made a great coat from an old horse blanket that gets  compliments in Paris and Milan.

 

Her definitions refer to clothing, but  could describe anything that becomes part of how we see ourselves culturally.  Recently some of us been talking about distinctions between fashion and style in mobile and wearable electronics down to their user experiences.  The social image you present using a mobile phone in public to the 'naturalness' of the user interface of a device.  There are interesting links to the increasingly divergent paths Google and Apple are taking.

 

Both companies have hired people associated with the fashion industry - Google for the Glass and Apple for a variety of positions.  Jheri works in the industry and is familiar with a few of the Apple hires by reputation.  She notes they deeply understand the difference between style and fashion.  A nimble company can add a fashion layer if they have a good control of design, but going deeper and focusing on style is more difficult.

 

An interesting emerging between the two companies is user privacy (not security, although that is related).³  Neither model is universally aligned with the needs of all users.  People say privacy is important, but most people don't place a high value on it and few understand its implications.⁴   The service they receive is sufficient compensation.  Apple is moving to a higher privacy model taking advantage of a natural alignment as their users are their customers.  The Apple ecosystem requires tight hardware and software integration and is perceived as valuable to many Apple customers - enough that they are willing to allow Apple to have high profit margins.  This division will become even more important as we connect to the things around us - it is just beginning to get interesting. 

 

Filters are a huge issue and are at the heart of our developing computer mediated interactions with each other, machines and information.  We are not having a robust public discussion on the filters stand between us and information and who controls them.  It isn't clear if we will have a discussion and if transparency and control will exist - even in small corners of our world.   This is a big thing - I don't know if it is the next big thing, but it has major implications.

 __________

¹ Trying to understand social filters and develop new filters and even relevant replacement filters is the job Google, FaceBook, OKCupid and other social platforms attempt.

² Style and fashion are often assumed to be the same.  Jheri and others in the industry make the distinction that appears in her definitions.  It would be too diverting to launch into a deeper discussion than the sketch applied here, but the sociology and implications for consumption are fascinating.  

³ A long discussion that would be diverting.  It is important to understand the distinction between security and privacy and  social, hardware and software architectures associated with privacy.

 ⁴ It's Complicated by danah boyd is an excellent introduction to teenage privacy constructs.

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recipe corner

 

Bagel Topology

 

Topology is the study of shapes.  If you have a rubbery coffee mug, you can reform it into a doughnut.  A flexible sheet can't be made into a sphere...  lots of fun things.  

 

It turns out you can cut a bagel and make linked halves.  

 

I learned about this in grad school and have made more than a few over the years.  The trick is to find a very solid bagel and use a good knife.  Make sure you don't stab yourself! - A friend was going to have her cello master's recital, but managed to stab herself cutting bagels during the reception before the concert.  She couldn't play for over a month.

 

This video is much better than any drawings I could make.  

on big data

The other day someone asked my opinion on Monsanto getting into agricultural "big data"..   People don't seem to agree on what that means, so I'll offer my definition, but first I should note that "data" means something different to someone in the physical sciences than the common usage.¹ 

 

"Big" is also loaded.  Big as compared to what?  I've seen datasets for some Fortune 50 companies that are trivially small compared to a week's worth of post trigger data from the LHC, but they are still sufficient to be overwhelming.  When I was a grad student in the 1970s my thesis experiment generated about 250 gigabytes a data and required a few years to analyze.  That was enormous at the time.  Our grant paid for several hundred thousand dollars worth of time on a supercomputer of the day that had less computational power than my iPhone 5.

 

Big data, for me, is dealing with the collection, processing and analysis of information that is at the frontier of what is possible.  

 

The devil is in the details and lurks at every step.  In the early 90s a few of us looked into the recording and billing system that AT&T used for domestic long distance calls.  This involved call detail records from 4ESS and 5ESS switches made by AT&T and DMS 100 and 200 switches made by Northern Telecom.  Call detail records are simple things - time stamps associated with the beginning and end of the call, some routing information, the type of call and so on.  Much of this had been cobbled together over several decades and too many assumptions had been made along the way.  We found error rates on the order of nearly two percent and were able to sort out quite a bit of what was going on in the system (the main technical report ran nearly 200 pages).  None of this had been properly characterized before and we underbelly missed bits and pieces, but we were able to characterize the system well enough to supply confidence levels on our measurements (we were a couple of physicists and a mathematician). 

 

Model building is extremely important.  You have to understand not only what you area measuring, but what you think it means.  This is can be extremely difficult in the social world and some huge leaps in logic are frequently made.  I've seen very robust work as well as work that is completely off base.  Sometimes there is enough information in your datasets to work with and sometimes other techniques are much more powerful.  Part of the success of Trader Joe's comes from being able to distill customer feedback that comes from their workers and be able to make good use of the information.  Big data analysis is not a big part of their strategy - it doesn't have to be as they are smart enough to make use of other more appropriate tools.

 

There is a danger associated with the misapplication of current and emerging tools.  Too often the underlying mechanisms are poorly understood and sometimes it is difficult or impossible to know what the tools are exactly doing.  The user may be well versed in their field but may not understand the problem at hand, details of the models being used, the quality of the data or several other critical components.  Compounding the problem is many of the newer tools is they have very powerful visualization components.  People tend to be more accepting and less skeptical of beautiful visual design. ...  lipstick for the pig

 

Too many organizations are jumping on the bandwagon simply because they think there is gold in their haystacks and, after all, isn't Hadoop cool?

 

Some have a solid understanding of their domain and can produce results that can be characterized.  I'm guessing there will be something of a bubble with a lot of less serious organizations getting burned in the next few years and a smaller number finding it is worth the serious effort to get things right.

 

Beyond that there are a few interesting things to note.  Particle physics is sometimes called the elder statesman of big data ... a lot of serious work understanding data collection, trigger analysis, modeling and analysis with results being compared with the work of others who were using their own techniques.   The complexity of the debris from a collision event increased with the collision energy of the accelerators and the resulting computational challenges have been matched (so far)  by a combination of Moore's Law and rapidly advancing algorithmic approaches.²

 

The Large Hadron Collider only keeps track of a tiny fraction of the data pre-selected by hardware and software based models called triggers that determine what is important. About three petabytes of data a month were kept going out to a network of several hundred thousand computers in three dozen countries.  The LHC is being modified for higher energies as is the computational environment.  When it turns on the data rates will increase significantly.  But even though this is a lot of data, it is relatively simple.  The events are isolated - there is no correlation between events.  Event correlation is a major challenge in astronomical surveys - the Square Kilometer Array, the Sloan Digital Sky Survey, the Dark Energy Survey and the Large Synoptic Survey Telescope come.

 

The LSST records at about 500 attributes of 20 or so billion objects in the sky.  A time series of these objects are recorded.  Attributes of these objects can change with time and there is a good deal of complexity associated with the early stages of capture and categorization - the transparency of the sky from night to night can dramatically change what something looks like.  Finding unexpected correlations in 500 dimensional spaces is at the edge and beyond what we can do at the moment ..   by my definition this is big data.³

 

A lot of research is going into exotic areas like compressed sensing, maximal information coefficient, and topological analysis and people are beginning to progress.  Some of it will be useful in other domains.

 

As a grad student I spent time with a Swede and a Californian hacking a fax machine to increase its resolution by a factor of about four.  We needed to communicate the sub and superscripts in our equations more clearly.  At the same time people were using the emerging computer internetworks to move computer typeset documents.  This became very common as the 80s went on, but order needed to be made of it.  There was a guy known as TimBL at CERN with an interesting proposal that he later implemented in the form of an HTTP client and server.  All to support the need for particle physics documents - usually time sensitive preprints - be be accessible by collaborations of groups around the world.  Sir Timothy Berners-Lee OM, KBE, FRS, FREng, FRSA had invented the web.

 

Mark Twain supposedly said something like "history doesn't repeat itself, but it does rhyme..."  I suspect folks doing the pure research in particle physics and astrophysics will create tools that will have an enormous impact on society.  

 

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¹ Not to go into detail, but information is the more basic term in physics.  Data generally has meta-information attached to it - how it was measured and to what accuracy for example.  Within the field it is assumed that someone has a good characterization of the meta-information when they invoke the term data.  

It should be said that physics is deeply into detail and understanding an apparatus, models and mathematical manipulations are of fundamental importance to the sport.

² Particle physicists learn about the very small by slamming particles into each other at nearly the speed of light.  The amount of energy during the collision is similar to early periods in the formation of the universe and allow the creation of exotic forms of matter. A friend notes that you wouldn't want to have one of us study a car or an airplane.  

³ The complexity is beyond linear it goes as 2^N, where N is the number of dimensions.  This is far faster than improvements from Moore's Law.  There are approaches to reason with this, but much remains to be done.

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Recipe Corner

 

A pressure cooker recipe.  If you don't have a pressure cooker and regularly cook, drop what you are doing and get one.  I did quite a bit of asking around and ended up with a Kuhn Rikon Duromatic and wouldn't hesitate to recommend it.  A bit spendy, but it is bulletproof and just works - an acceptable price for one of the most important pieces of kit in my kitchen.

 

Red Lentils and Jasmine Rice

 

Ingredients

 

° 2 tsp olive oil 

° 1 tbl minced ginger

° 2 cloves garlic, minced

° 2 cups jasmine rice (I used a brown jasmine)

° 1/2 c red lentils (you could use other varieties)

° 3 cups water or vegetable broth (really rich if you have a good broth around, but ok with water)

° sea salt 

 

Technique

 

° heat the cooker over medium heat, add the oil.  Sauté the garlic and ginger for about a minute.  Stir in the rice, lentils and add the fluid.

° lock the lid and bring to pressure.  Lower the heat to maintain pressure and let it cook for 18 minutes.

° move pot off the burner and let it come to atmospheric pressure naturally

° season to taste (a bit of salt for me)

 

paths past 269

Just a quick post as I've been without net, electricity and heat for the past week.  I finally have the modern conveniences and it happens to be election day, so a quick post.  There are a few other posts that should come in the next week as I had a lot of time to think.

 

Today votes are cast for the election of the President of the United States.  Technically voters don't directly elect the president, but rather 538 electors.¹  To be elected President a candidate needs 270 or more electoral votes.

 

There are dozens of polls all with some noise.  Some are designed and executed better than others, some have built-in biases and most tend to move around during a campaign.  The press tends to focus on them and take them as the odds that a candidate might win or lose the race.

 

Recently Nate Silver, author of the 538 blog, has been criticized for suggesting that President Obama currently has a better than 90% chance of being re-elected despite the likelihood that the election might be very close - possibly a one percent or less popular vote margin for the winner.  This strikes some as entirely daft - how could the President have such a good chance in such a "tight" race?

 

It turns out this is a nice example of a statistical approach.  There is no inconsistency with a close election and a relatively high probability of a certain result - in this case the result required for a win is the accumulation of 270 or more electoral votes.

 

Rather than rely on one or a small number of national polls, the trick is to take a group of state polls and calculate a total probability that a candidate achieves 270 or more in a combination of ways.  Silver has a model that has some assumptions about the past accuracy and predictive properties of each of the individual polls as do others who use this approach. 

 

There are many possibilities to sort through - something over 2.3 quadrillion.  No one would try to be exact and some who use this meta analysis of polls look at a few thousand possibilities that are usually chosen to look at the important "battleground" states.²  

 

It should be noted that the various meta-analysis approaches are currently giving Obama anywhere between a 70% and 90% of keeping his job.  

 

We tend to have a notion that the popular vote is important, but to first order it isn't.  There is a weighting forced by the electoral college - the game that really matters - and districting.

 

It is easy to make some wrong assumptions about the real problem, add our own biases (and there are many with politics!) and slip into innumeracy.  I'm reminded of a section from John Paulos' book  Innumeracy

  

If from some stock market advisor you received in the mail for six weeks in a row correct predictions on a certain stock index and were asked to pay for the seventh such prediction, would you? Assume you really are interested in making an investment of some sort, and assume further that the question is being posed to you before the stock crash of October 19th, 1987. If you would be willing to pay for the seventh prediction (or even if you wouldn't), consider the following con game.

Some would-be advisor puts a logo on some fancy stationery and sends out 32,000 letters to potential investors in a stock index. The letters tell of his company's elaborate computer model, his financial expertise, and inside contacts. In 16,000 of these letters he predicts the index will rise, and in the other 16,000 he predicts a decline. No matter whether the index rises or falls, a followup letter is sent, but only to the 16,000 people who initially received a correct "prediction". To 8000 of them a rise is predicted for the next week, to the other 8000 a decline. Whatever happens now, 8000 people will have received two correct "predictions". Again to these 8000 people only, letters are sent concerning the index's performance the following week, 4000 predicting a rise, 4000 a decline. Whatever the outcome, 4000 people now have received three straight correct predictions.

This is iterated a few more times until 500 people have received six straight correct "predictions". These 500 people are now reminded of this and told that in order to continue to receive this valuable information for the seventh week they must each contribute $500. If they all pay, that's $250,000 for our advisor. If this is done knowingly and with intent to defraud, this is an illegal con game. Yet it's considered acceptable if it's done unknowingly by earnest, but ignorant publishers of stock newsletters, or by practitioners of quack medicine, or by television evangelists. There's always enough random success to justify almost anything to someone who wants to believe.

There is another quite different problem exemplified by these stock market forecasts and fanciful explanations of success. Since they're quite varied in format and often incomparable and very numerous, people can't act on all of them. The people who try their luck and don't fare well will generally be quiet about their experiences. But there'll always be some people who will do extremely well, and they will loudly swear to the efficacy of whatever system they've used. Other people will soon follow suit, and a fad will be born and thrive for a while despite its baselessness.

There is a strong, general tendency to filter out the bad and the failed and to focus on the good and the successful. Casinos encourage this tendency by making sure that every quarter that's won in a slot machine causes lights to blink and makes its own little tinkle in the metal tray. Seeing all the lights and hearing all the tinkles, it's not hard to get the impression that everyone's winning. Losses or failures are silent. The same applies to well-publicized stock market killings vs. relatively invisible stock market ruinations, and to the faithhealer who takes credit for any accidental improvement, but will deny responsibility if, for example, he ministers to a blind man who then becomes lame.

 

I'm a huge believer in basic statistics and probability courses in high school - it is at least as important as basic calculus for modern citizens to sort through information available and not be so inclined to take predictions on face value.  It is critically important to understand as much as possible about information and how it is filtered and manipulated.  There is real danger in considering the thing some call "data" as pristine and reliable.

 

Back to the election.  Many segments of the universe of voters are extremely stable and elections can be influenced by encouraging or suppressing some of them.  Black and Hispanic voters tend to vote for Democrats, so elections with poor turnout can pivot favorably for Republicans and the urge for that party to suppress votes has to be strong.  On the other hand the Democrats would want to focus on voter registration and get-out-the vote efforts in districts where a strong turnout can make a different.  The same can be said about fundamentalist Christians voting for Republicans and there are a huge number of segments of varying degrees of strength and reliability.  All of this gives enormous opportunities for trying to datamine and produce lists so these potential voters can be reached and potentially influenced.  

 

It will be interesting to see how the meta-analysis folks make out this time.  They usually do better than those who analyze (and mostly over-analyze) popular vote polls.

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¹ The number is based on the membership of the US Congress: 435 House members, 100 Senators, and 3 electors from the District of Columbia.  

² There are different approaches.  The Princeton Election Consortium uses a technique that looks at the probability of getting an exact number of votes.  It turns out you can do this by calculating a relatively simple polynomial expression.  To get a deeper look at a daily prediction from the Princeton group check this out.

 

 

 

 

don't even bother trying to comb a hairy banana and a gathering of themes

Or a hairy ball for that matter.. at least not if you want to perfectly comb them so their hair lies perfectly flat everywhere on their surface.

 

Imagine you have an unruly donut hair covered donut and want to comb it flat (I can't imagine eating a hair covered donut).  It turns out to be an easy task.  Now think about the problem of combing a hair covered ball.  It turns out you can't do it perfectly - the best you can do is to make the fur like flat (locally tangent to it's surface) at every point but one where you are left with a cowlick. 

 

\ A banana is really the same thing as a sphere topologically.  Imagine the banana was incredibly pliable - really really ripe - and the skin just as flexible.  You could push in bits here and pull out others there and change its shape into a sphere.¹

 

The hairy ball theorem has been proven.² Math, unlike science,  is an areas where solid black and white proofs are possible.

 

It turns out there is always at least one point on the planet where there is no wind.  This point is the eye of a cyclone or anticyclone (wind spirals around it - air can't flow into or out of it as it is static).  So there is, and has been, at least one cyclone everywhere on Earth ever since there was an atmosphere.  You can say the same about any body in the Universe with an atmosphere.

 

so if you want to take bets ...

 

And if you wonder what brought this on, the banana I had for breakfast this morning was on the verge of being too ripe and looked like it might be growing hair one of these days.  I also have an unruly cowlick today - put this together and some math has to come out.  Math is neat stuff!

 

While I'm far from being fashionable, some of you play it well.  It is really fascinating to think about as there is an ephemeral nature and personal style can be very artistic.  In the US personal style for many has been replaced by inexpensive, ephemeral uniforms.  Getting something cheap that looks like something that a vetted style setter wears has become more important than finding a personal style.  In theory there is much more choice in stores, but in reality this translates into a very narrow "look".  

 

Jheri and I have been re-thinking our views on mass customization.  The genesis of this was marrying some work I did for the Defense Department with the need of Jheri and some others to find clothing that fits.  The notion was that mass customization was becoming easier and fit was the most important component.  While fit still is the most important component there are clearly other drivers as well as roadblocks.  Recently Jheri pointed to a new book on overly-cheap apparel (halfway into this post).  The book is mostly about the over consumption of very low quality fashion and the dramatic change that has swept the industry.  We are now wondering if some changes we're seeing among the 20 to 30 year old demographic may be important sources of kindling for the ignition of real mass customization - forget things like fancy printing technologies which are still a long way off from large scale viability ...

 

Apparel is still largely handmade at this point and the cost structure is going to keep it that way for at least another decade or two.  A pair of jeans may have a few hours of hand labor in them, but if you can do that for two dimes an hour, why change? Major designers are now working for the high volume firms, but the quality of material and construction is poor.  The middle tier of quality and manufacture has mostly disappeared.  A few speciality makes have appeared that exploit this need - Lululemon is a good example.³  Their quality matches their pricing, but this is fairly rare - to first order even $300 dresses are very poorly made.  Many consumers no longer have the ability to judge construction and material quality.

 

The signal we are seeing among at the edge - mostly in places like San Francisco, NYC (especially Brooklyn and lower Manhattan!), Seattle, Minneapolis and San Diego may be the rejection of overconsumption and more value placed on owning a smaller number of higher quality items that can last.  These are also areas that tend to support the notion of a strong sense of personal fashion.  There is more of the diversity you see in some European cities and mall uniforms are distinctly clueless.

 

New small business is emerging to fill these needs and some people are becoming craftsmen and artisans.  Clothing repair and re-tailoring shops are exist in greater number and I even saw a few new cobbler shops in Brooklyn last week.   While it isn't clear that this is a trend that will spread, it should be pointed out that the traditional apparel industry control on media is much weaker with forces like Pinterest exploding in popularity and Etsy creating a new channel for accessories.  

 

It is very curious to note that those who ply vintage stores note that items at least thirty years old are worthy of salvage and modification - newer items become progressively more poorly made to the point where anything made in the past decade is so poorly made that it isn't worth bothering with.

 

There are some great opportunities for design tools and even manufacture automation that may come to the small scale shops before the large manufacturers.  The big guys (like Levis) have tried what they thought was mass customization a decade ago and have mostly abandoned it.  I think this is like Microsoft's decade old notion of the tablet - they simply lacked the chops to dive in deep and re-imagine that was really needed. TRVL is another example of a publication more properly reimagined for the iPad world - and a need to break beyond conventional tools (like InDesign and pdfs used by the publishing industry) and thinking.  They were trying to hammer their square peg into a round hole and, guess what... it didn't work very well.  

 

It is likely this won't be very large at first - a ten percent shift in the market within this age segment would be big in my mind, but very indicative of a major trend.  We may see the re-emergence of the middle tier of apparel quality and manufacturing along with people taking a more focused interest in what fashion is - they may become their own artists rather than mere consumers.⁴  Remember - these are the guys who see iPads and creative tools.  My generation tends to see them (and very improperly!) as consumption devices.

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¹ Try a little experiment:  The famous example would be to change a donut into a coffee mug - cookie dough will show the way and had a big advantage over modeling clay in that it is edible. This can make for a sweet introduction to topology:-)

² There are many proofs. Most are pretty complex requiring some deep math - usually homology theory is involved.  I could follow some in the day, but now I'm more rusty - you have to keep at this to be in good form.  I did see an elegant and simple proof by John Milnor when I was a student and have it in my notes.  Since at least one of you is a mathematician and several are physicists and serious computer scientists it is reasonable to show it.  Normally it would take at least an hour to typeset it in LaTeX and arrange images to embed in html, but here is a nice reproduction in a math blog. The warning is this is probably greek if you haven't done an college level math - so please ignore it and accept this a proven result, but if you had an advanced calculus course, you have the tools you need and it is a sweet little proof.

³ I've visited their shops a few times.  Once with Jheri trying to learn a bit and then out of curiosity when I was in Manhattan.  Several of the customers told me construction and material quality is much more important than the very careful marketing - it represents good value and is what makes them repeat customers.  The chain has established itself as one of the few mid-quality outlets in existance.  Sort of a sad thing when mid quality is the highest level most of us have access to - and many of us can't even get that.  A real opportunity space...

⁴ Distributed design and IP ownership needs tools that are probably closely linked with social media.  Improved mathematical models to convolve design and fit patterns need to be made along with improvements in user/designer feedback.  Automated and semi-automatic manufacturing tools.  More uniform measurement tools, improvements in user and designer feedback, user and designer education tools ...  the list goes on and on and the surface has only been scratched.  But it is important to note that this is an enormous industry - something like a half trillion a year in the US.  Even niches can be non-trivially large.

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Before getting on with the recipe, I'd like to make a recommendation.  I'm afraid my education is very narrow.  High school was ok, but in college I focused and spend most of my time in the physical sciences and math - placing out of other subjects where possible.  A really big mistake in retrospect.  Then on to grad school with a much more intense specialization.    

 

I do read a lot and am reasonably curious, but there isn't a lot of structure to this approach.  Recently I've been taking a few courses on iTunes U (there are other flavors of this sort of online learning). I'm not diving in deeply and doing the homeworks and all of the readings at this point, but am enjoying those with great lectures.  It is an excellent way to pass time on long boring trips and during exercise sessions.  Over a three week period, for example, I listened to all of the lectures in David Blight's excellent History 119 course at Yale: The Civil War and Reconstruction Era. Since then I've read a few of the books on his reading list.  Great stuff and highly recommended!

 

All of the iTunes U online material is free and some of the classes are extremely interesting.  It isn't a substitute for deeply focusing and doing all of the coursework and interacting with the professor, but it is so much better than TV or movies.

 

Here is a guide I use to hunt courses.  I'm sure there are others - if you know of good ones, let me know.  And a recommendation for me - an excellent short introduction to Quantum Mechanics from Oxford by James Binney.  He does an excellent job.  The warning is you really need to work out the problems to gain any understanding of the subject.  The class notes, recommended books and problem sets are here.

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Recipe section

 

Yesterday saw 97°F here and the urge for a homemade milkshake was strong as recently I had a rather poor excuse from what had been described as the best ice cream parlor in the area.

 

Homemade tahitian vanilla bean ice cream, homemade chocolate sauce + broken bits of Lindt 72% (added after the blending).  I used my antique Hamilton Beach milkshake maker from the mid 1960s - a very high speed spindle and careful attention is the trick for velvety textured, very thick shakes.  This one was - well - perfect (sigh)……

 

I won't go through this one exactly, but rather give a cheater's version that may be just as good for you.

 

 

Chocolate-Chocolate Chip Milkshake

Ingredients (for one or two servings depending on how much you like milkshakes)

° a pint of super premium vanilla ice cream - I like Häagen-Dazs Five Vanilla Bean as it is reasonably well crafted, widely available and is very simple with only five ingredients. Depending on where you are you may be able to find better and a really excellent homemade can be far better.

° about 120g whole milk (a half cup)

° a good chocolate topping - I've been making my own and can't make a recommendation.  I've seen many in stores.

° 20-40g dark chocolate chopped into small bits - I used Lindt 72% 

 

Technique 

° For a thick and velvety shake you really need a high speed spindle.  I use an ancient Hamilton Beach single spindle unit from the 1960s.  It's shaft runs at 17,000 rpm and should be treated with care.  For a consistency like many current ice cream parlors use a blender, and for a very thick shake find a study container for the mixture and use a table knife and large, heavy spoon and a LOT of elbow grease.

° cool the milk to near freezing and let the ice cream sit on your countertop for about 3 or 4 minutes before starting to soften it.

° Mix the milk, ice cream and chocolate syrup and blend using one of the techniques above.  I start with eight parts of ice cream for every part of milk - if you like them to be drinkable use a four to one mixture.  If you are using a pint of ice cream, try about 2 table spoons of chocolate syrup for a very light chocolate and increase as desired.  I like the lighter version.

° After blending mix in the chocolate chunks with a table knife.

° Pour into a chilled glass and, if you like, top with a bit of whipped cream and/or shaved chocolate.  I couldn't take it anymore and just had it as is.