Monday, March 30, 2015

Notes on Mario Livio's book, The Golden Ratio

_The Golden Ratio:  The Story of Phi, the World's Most Astonishing Number_ by Mario Livio
NY:  Random House, 2002
ISBN 978-0-7679-0815-3

(6)  Roger Herz-Fischler, _A Mathematical History of the Golden Number_

(26)  Pythagoras emphasized the importance of learning above all other activities, because, in his words, "most men and women, by birth or nature, lack the means to advance in wealth and power, but all have the ability to advance in knowledge."

(67)  The key figure and driving force behind the geometrical theorems concerning the Golden Ratio was probably Theaetetus (ca. 417 BC - ca. 369 BC), who according to the Byzantine collection _Suidas_ "was the first to construct the five so-called solids."
NB:  Which actually go back to at least the Neolithic

(79)  In other words, in a regular pentagon, the ratio of the diagonal to the side is equal to ø.  This fact illustrates that the ability to construct a line divided in a Golden Ratio provides at the same time a simple means of constructing the regular pentagon.  The construction of the pentagon was the main reason for the Greek interest in the Golden Ratio. The triangle in the middle of Figure 25a, with a ratio of side to base of ø, is known as a Golden Triangle;  the two triangles on the sides, with a ratio of side to base of 1/ø, are sometimes called Golden Gnomons.

(81)  The Golden Ratio has the unique properties that we produce its square by simply adding the number 1 and its reciprocal by subtracting the number 1.  Incidentally, the negative solution of the equation x sub 2=(1-√5/2) is equal precisely to the negative of 1/ø.

(85)  The Golden Rectangle is the _only_ rectangle with the property that cutting a square from it produces a similar rectangle.
NB:  Successive cutting results in the ability to trace a logarithmic spiral converging on one point, the so-called Eye of God

(101)  As we go farther and farther down the Fibonacci sequence, the ratio of two successive Fibonacci numbers oscillates about (being alternatively greater and smaller) but comes closer and closer to the Golden Ratio.

(111-112)  One of the discoveries of the Bravais brothers in 1837 was that new leaves advance roughly by the same angle around the circle and that this angle (known as the divergence angle) is usually close to 137.5 degrees.  Are you shocked to hear that this value is determined by the the Golden Ratio?  The angle that divides a complete turn in a Golden Ration is 360º/ø=222.5 degrees.  Since this is more than half a circle (180 degrees), we should measure it going in the opposite direction around the circle.  In other words, we should subtract 222.5 from 360, giving us the observed angle of 137.5 degrees (sometimes called the Golden Angle).

(126)  Three of Piero's [della Francesca] mathematical works have survived:  _De Prospective pingendi (On perspective in painting), _Libellus de quinque Corporibus Regularibus (Short book on the five regular solids), and _Trattato d'Abaco (Treatise on the abacus).

(140)  Dürer's polyhedron in Melancolia I:

(146-147)  Kepler:  The Earth's sphere is the measure of all other orbits.  Circumscribe a dodecahedron around it.  The sphere surrounding it will be that of Mars. Circumscribe a tetrahedron around Mars.  The sphere surrounding it will be that of Jupiter.  Circumscribe a cube around Jupiter.  The surrounding sphere will be that of Saturn.  Now, inscribe an icosahedron inside the orbit of the Earth.  The sphere inscribed in it will be that of Venus.  Inscribe an octahedron inside Venus.  The sphere inscribed in it will be that of Mercury. There you have the basis for the number of the planets.

(155)  Kepler's songs of the planets: and in modern notation at

(171)  Another art theorist who had great interest in the Golden Ratio at the beginning of the twentieth century was the American Jay Hambidge (1867-1924).  In a series of articles and books, Hambidge defined two types of symmetry in classical and modern art.  One, which he called "static symmetry," was based on regular figures like the square and equilateral triangle, and was supposed to produce lifeless art.  The other, which he dubbed "dynamic symmetry," had the Golden Ratio and the logarithmic spiral in leading roles. Hambidge's basic thesis was that the use of "dynamic symmetry" in design leads to vibrant and moving art.  Few today take his ideas seriously.

(193)  [Joseph] Schillinger was  a great believer in the mathematical basis of music, and, in particular, he developed a System of Musical Composition in which successive notes in the melody followed Fibonacci intervals when counted in units of half-steps.

(205)  One of the most startling properties of any Penrose kite-dart tiling design is that the number of kites is about 1.618 times the number of darts.  That is, if we denote by Nkites the number of kites and Ndarts the number of darts, then Nkites/Ndarts approaches ø the larger the area we take in....

Another pair of Penrose tiles that can fill the entire plane (nonperiodically) is composed of two diamonds (rhombi), one fat (obtuse) and one thin (acute).  As in the kite-dart pair, each of the rhombi is composed of two Golden Triangles or Golden Gnomons, and special matching rules have to be obeyed (in this case described by decorating the appropriate sides or angles of the rhombi) to obtain a plane-filling pattern.  Again, in large areas there are 1.618 times more fat rhombi than thin ones, Nfat/Nthin=ø.

(206)  Penrose's work on tiling has been expanded to three dimensions.  In the same way that two-dimensional tiles can be used to fill the plane, three-dimensional "blocks" can be used to fill up space.  IN 1976, mathematician Robert Ammann discovered a pair of "cubes", one "squashed" and one "stretched," known as rhombohedra, that can fill up space with no gaps.  Ammann was further abel to show that given a set of face-matching rules, the pattern that emerges is nonperiodic and has the symmetry properties of the icosahedron; this is the equivalent of fivefold symmetry in three dimensions, since five symmetric edges meet at every vertex).  Not surprisingly, the two rhombohedra are Golden Rhombohedra - their faces actually are identical to the rhombi of the Penrose tiles.

(212)  Consider the following simple algorithm for the creation of a sequence known as the Golden Sequence.  Start with the number 1, and then replace 1 by 10.  From then on, replace each 1 by 10 and each 0 by 1.
NB:  What is this in binary?
181 181 21
181 22

(216)  These musings have turned into the by now-famous question:  "How long is the coast of Britain?"  Mandelbrot's surprising answer is that the length of the coastline depends on the length of your ruler.
NB:  Zeno's paradox

(219)  Clearly, for many systems (eg, a drainage system or a blood circulatory system), we may be interested in finding out at what reduction factor precisely do the branches just touch and start to overlap.  Surprisingly (or maybe not, by now), this happens for a reduction factor that is equal precisely to _one over the Golden Ratio_, 1/ø=0.618...  This is known as a Golden Tree, and its fractal dimension turns out to be about 1.4404.
NB:  Constructal theory

(224) Elliot's basic idea was relatively simple.  He claimed that market variations can be characterized by a fundamental pattern consisting of five waves during an upward ("optimistic") trend and three waves during a downward ("pessimistic") trend.

(233)  Newcomb... came up with an actual formula that was supposed to give the probability that a random number begins with a particular digit.  That formula gives for 1 a probability of 30%;  for 2, about 17.6%;  for 3, about 12.5%;  for 4, about 9.7%;  for 5, about 8 %;  for 6, about 6.7%;  for 7, about 5.8%;  for 8, about 5%; and for 9, about 4.6%.

Wednesday, March 25, 2015

BCSEA Webinar: Full Charge on Electric Cars!

webinar:  British Columbia Sustainable Energy Association
webinar video at

John Stonier, VeloMetro

Up to 1915, about half of the cars on the road were electric
Edison's nickel steel batteries in at least one electric vehicle have lasted until 1990
300,000 electric vehicles in USA, mostly in WA, OR, CA
The market has recently switched to buying all electric rather than hybrid
8 states have zero emission vehicle mandates:  CA MA CT OR Maryland RI NY and VT
Electricity is about 20¢ per liter equivalence to gasoline [in British Columbia]
Range is now sufficient for in-city but not quite yet for between cities driving but 300 kilometer [186 mile] batteries are in the pipeline
Hydrogen fuel cell cars are more expensive and more expensive to operate than electric 
An electric car is by definition a luxury vehicle and will last longer with lower operation costs:  1/6 operating costs, 1/6 maintenance, 1/3 depreciation, 3-5x life expectancy of gasoline vehicels
Batteries for Nissan Leaf should last 15-20 years
20% less total costs than internal combustion (his calculation is more like 40% over 10 years)
Tipping points:  condo charging for urban drivers (load sharing), access to HOV lanes, higher incentives, inter-city charging strategy, public charging for hospitality industry - EV tourism in BC, distance based auto insurance
Three DC fast charge standards:  CHAdeMO, SAE Combo, Tesla - SAE may be best
More light truck options:  Nissan NV200EV, Smith Electric, EV Fleet makes pickup sized vehicles, Via Motors makes the VTrux
Autonomous vehicles have all been electric
Vehicle to grid - decentralize storage and demand response

Q: battery upgrades?  Tesla is offering one for earlier Roadsters.
Yes but Nissan has changed the size of the battery bay which may require chemistry advances and may be some aftermarket suppliers too.
Q:  Critical business models?
Tesla did it right - start with luxury for quicker returns, attracting influencers and early adopters, corporate sales and direct marketing
Q:  Five person car?
Nissan and Tesla both have family sized cars and utility vans (Nissan).  Not yet light trucks.
Q:  Pull a trailer?
Hybrids also don't pull trailers.  Not sure why
Q:  Coal burning for electricity?
Electric cars use 1/4 of the energy to do the same thing and the grid is getting greener.  Coal is more costly than alternatives.

VeloMetro trial is happening in Vancouver this summer.  Human power, solar electric hybrid vehicle, less than 200 pounds.

Sunday, March 22, 2015

City to City: Mobility - Iceland to Boston


Over 60 communities in Iceland district heated, moving to electrification of transportation rather than hydrogen economy now.

Vern Global:  100% renewably powered data center. Iceland geothermal and with wind at 40% efficiency as opposed to 7% in EU, the second most reliable grid in the world (after Germany?)

Dagur Eggertsson, mayor of Reykjavik:  first public school connected to district heating in 1930 and whole city by the 1970s, now also producing electricity.  Coal phased out by 1967.  23% of energy use in transportation and produces most of their ghgs.  As much a car culture as US.

David Keith, Sloan School, studying adoption of alternative fuel vehicles.  Turnover of vehicle fleet is painfully slow, each vehicle lasts about 16 years in Iceland. New tech adoption is socially contagious - the more sold, the more will be sold.

Vineet Gupta, Boston city transportation:  1/3 of the city is 18-45, 60% of all trips are by foot or bike.  4000 electric vehicles in MA. Half of the public land in Boston are streets and sidewalks.  Infrastructure for on-demand transportation.

Ryan Chin:  electrification, sharing systems, autonomy for transportation.  Singapore is a case study. 300,000 cars could provide the whole 5.4 million people with transport within 20 minutes as shared vehicles.
Bjorgvon Sigurdsson, national electric company of Iceland, Landsvirkjun - all of the electricity produced in Iceland 100% renewable and largest producer of electricity per capita (Norway is second).  Producing methanol as green fuel.

Wolfgang Gruel, Daimler:  car2go - easy car sharing system, now in 29 cities in 8 countries.  Moovel - a platform to simplify mobility, map your trip and buy tickets for the journey.

Einar Gudmundsson, Arion Bank:  business accelerators - Startup Reykjavik and Startup Energy, 54 startups so far.

Q: how to manage the transition?
Small experiments that can fail until something works.

RC: energy will be more distributed and microgrids
DE:  competitiveness is all about the quality of life