The AFL though the lens of Cherin-Gordon

To an Australian encountering US team sports, and soccer in Europe, it would no doubt feel surprising to discover how many games are played in Major League Baseball, the National Hockey League, and the National Basketball Association. Whereas no major sport in Australia plays more than 25 games per season, those three leagues play 82 games (NBA and NHL) or 162 games (MLB). European soccer leagues play between 30 and 46 games per season.

In his Untangling Skill and Luck from 2012, Michael Maubossin noted that a 50 percent ratio of skill to luck can be achieved by:

• 12 NFL games
• 69 MLB games
• 15 NBA games

For ice hockey and soccer, estimates are extremely variable, ranging from 35 to 75 games for ice hockey and 25 to 50 for soccer.

Last year – though I discovered this only today – journalist Simon Cherin-Gordon offered a further insight into the difference in season length between gridiron (NFL) and the other major leagues in North America. In an article titled ‘Why Are We Playing 82 Games, Anyway’ – written to advocate a shorter NBA season, which Maubossin had shown to be more practicable than any other US or European team sport – Cherin-Gordon said:

“Much of this has to do with the inherent nature of each sport. A baseball star will only bat four or five times in an average game, or pitch one out of every five days. Runs are hard to come by, and lower scores lead to higher variance and make a short season untenable (the same can be said for the NHL, which has an 82-game season just like the NBA).” 

“[Gridiron] Football, meanwhile, is a sport where the best players are involved in somewhere between 60 and 70 snaps per game (which constitutes roughly half of the game’s plays). Its scoring is less random, and the best team usually wins. It is also as violent a sport as there is, thus frequent games and a long season are out of the question” 

“While the NBA season falls just in between that of the MLB and NFL in terms of length, it has far more inherent similarities to [gridiron] football than it does to baseball.” 

“If you think 60-70 plays per game is a lot for a [gridiron] football player, try more than doubling that. Basketball players play both offense and defense, and NBA games generally feature around 200 possessions. The best players play between two-thirds and three-quarters of those possessions (32-36 minutes).” 

“That’s just one of many ways the NBA takes out variance. It is the only sport where every successful play is quantified on the scoreboard. There are no base hits or first downs. When you do score in other sports, it is a more random occurrence with more impactful ramifications. Crossing home plate or the goal line one time is worth exponentially more than putting the ball in the hoop. There are also less [fewer] players involved, and less [fewer] quirky advantages to playing at home. The best team in an NBA game wins more often than they do in any other sport.”

What strikes me about this list is that if we apply what Cherin-Gordon says to the AFL, we have the following comparison:

Sport	“Possessions” per game	% played by best players	“Possessions” played by best players
Baseball	≅240 (pitches)	≤⅙ (16.67 percent)	≤40 (pitches)
Gridiron	≅140 (downs)	≅½ (50 percent)	≅70 (downs)
Basketball	≅200	≅¾ (75 percent)	≅150
(Australian rules) football	≅600-700 (total possessions)	≥⅞ (87.5 percent)	≥525

This table does not measure how many team possessions there are per (Australian rules) football game. However, in basketball the rules make teams to alternate possession after every score, which gives weaker teams more of a chance than in football, where a team can extremely easily score and then regain possession without one opposition possession. (In one game in 1954, it is known Richmond kicked five goals without a Melbourne player touching the ball, and as the Demons actually won that game this is unlikely to be the most extreme possible case.) Football is also analogous to basketball in that every “successful” play is rewarded on the scoreboard – although if a team kicks out of bounds on the bounce or a pack forms that team can regain possession without an opponent necessarily touching the ball.

The table above suggests that, although the AFL season is much shorter than those of baseball, ice hockey or soccer, it is not proportionately so relative to the requisite length for a passable skill-to-luck ratio. In fact, if in Cherin-Gordon’s words,

“MLB needs 162 [games], the NHL needs 82 [games]... the NFL needs 16 [games]”

then the AFL would “need” no more than 10, or maximally 12, games per season.

However, although Australian football has exceedingly low internal variance via randomness in player performance, external variance from variation in weather and ground conditions was highly significant before the climatic “magic gate” of 1997/1998 and the closure of Waverley Park and the AFL’s old suburban grounds. Subsequently this external variance has been effectively eliminated by:

1. rapid poleward expansion of the subtropical arid belt, totalling 800 kilometres since 1964, which has:
• reduced the incidence of wet weather and increased evaporative drying of grounds when it does rain
• also reduced the possibility of extremely windy conditions that favour shorter players
2. artificial drying of grounds during rare cases of wet weather, and improved drainage
3. the opening of closed-roof Docklands Stadium where weather conditions are consistently dry

This history of significant external variance undoubtedly explains why the AFL has a longer season than required for a passable skill-to-luck ratio. In addition, on softer surfaces football was less physically taxing than the present-day game, as seen by players frequently playing mid-week Foster’s Cup matches as well as weekend premiership matches. More than that:

1. even a minor reduction in games would be extremely unpopular with football fans, as demonstrated in the 20-game 1993 season
2. a reduction to 10 or 12 games would require a radical restructuring of the AFL’s season
3. many pairs of teams would – as in the NFL between 1978 and 2001 – go up to twenty seasons without opposing each other
4. it is highly plausible that the game would become even more vigorous and physically demanding with only half as many games, so that the reduction in injuries would be much less than Cherin-Gordon supposes (this also applied to the NBA)

Most likely, only major increases in severe – likely only in career-ending – injuries would cause the AFL to reduce the number of games on its schedule. This is plausible in an even hotter and drier climate than observed today thanks to Australian and Gulf Cooperation Council greenhouse pollution. However, if a shorter season be judged inevitable it could create a vicious circle of harder, more anaerobic play and more injuries, because if AFL players knew they had only 10 or 12 home-and-away games to play, they would almost certainly play harder than they do now knowing they have to get through 22.

Rarity of full-period prime denominators in rational approximations of irrational numbers

Ever since, a few months ago, I began to discover that – in addition to the very well-known approximation of 22/7 for 𝜋 – other irrational numbers had frequently used rational approximations, I have been struck by one fact: that full-period primes occur very rarely as denominators in such fractions, especially in the most useful rational approximations for hand calculations.

In order to test this hypothesis I have compiled a representative selection of irrational numbers in the table below, useful rational approximations for these numbers, and the periods of these rational approximations. For most numbers the most common rational approximation has been used; in certain cases like the square root of 6 and 𝜋, I have given more than one rational approximation, with that with the larger denominator naturally more accurate.

Number	Decimal expansion	Rational Approximation	Period	Prime factorisation of denominator	Character and type of prime factors
√2	1.4142135623730950488016887242	99/70	6	2•5•7	Composite Full-period and terminating factors
√3	1.7320508075688772935274463415	97/56	7	2•2•2•7	Composite Full-period and terminating factors
√5	2.2360679774997896964091736687	161/72	1	2•2•2•3•3	Composite Short-period (unique) and terminating factors
√6	2.4494897427831780981972840747	49/20	0	2•2•5	Composite Terminating decimal
		218/89	44	89	Half-period prime
√7	2.6457513110645905905016157536	127/48	1	2•2•2•2•3	Composite Short-period (unique) and terminating factors
√10	3.1622776601683793319988935444	117/37	3	37	Short-period (unique) prime
√11	3.3166247903553998491149327366	199/60	1	2•2•3•5	Composite Short-period (unique) and terminating factors
∛2	1.2599210498948731647672106072	63/50	0	2•5•5	Composite Terminating decimal
∛3	1.4422495703074083823216383107	75/52	6	2•2•13	Composite Half-period prime factor
∛4	1.5874010519681994747517056392	100/63	6	3•3•7	Composite Half-period and short-period (unique) factors
		227/143	6	11•13
∛5	1.7099759466766969893531088725	171/100	0	2•2•5•5	Composite Terminating decimal
∛6	1.8171205928321396588912117563	467/257	256	257	Full-period prime (accurate to 1-in-33,629,323!)
∜2	1.1892071150027210667174999705	44/37	3	37	Short-period (unique) prime
∜3	1.3160740129524924608192189017	25/19	18	19	Full-period prime
		229/174	28	2•3•29	Composite Full-period, short-period and terminating factors
21/5	1.1486983549970350067986269467	85/74	3	2•37	Composite Short-period (unique) and terminating factors
21/12	1.0594630943592952645618252949	89/84	6	2•2•3•7	Composite Full-period, short-period and terminating factors
𝜋	3.1415926535897932384626433832	22/7	6	7	Full-period prime
		355/113	112	113	Full-period prime
e	2.7182818284590452353602874713	193/71	35	71	Half-period prime
ee	15.154262241479264189760430272	197/13	6	13	Half-period prime
		2849/188	46	2•2•47	Composite Full-period prime factor
e𝜋	23.140692632779269005729086367	1481/64	0	2•2•2•2•2•2	Terminating decimal
ln 2	0.6931471805599453094172321214	61/88	2	2•2•2•11	Composite Short-period (unique) prime factor
log10 2	0.3010299956639811952137388947	59/196	42	2•2•7•7	Composite Terminating and squared full-period prime factor

If we study this table, we see that, for whatever reason, there seem to be very few full-period prime denominators. The two well-known approximations for 𝜋, one approximation for the fourth root of three, and one remarkable approximation for the cube root of six are the only exceptions. [The cube root of six – which has minor notability as the geometric mean of 1, 2 and 3 – I did not originally intend to include but decided to do so because the approximation ⁴⁶⁷/₂₅₇ is so amazingly accurate, being superior to the famous Milü approximation for 𝜋].

Why this should be so is an interesting question. It is possibly because the way in which the continued fractions used to find such approximations as ⁴⁶⁷/₂₅₇ for ∛6 would add factors in the finding of “common denominators” needed for addition of fractions, although I have not checked this yet.

A final proof the “March on Canberra” is a quarter-century and counting overdue

According to a new paper in the journal Nature, regardless of what the rest of the world does, record-breaking temperature rises are already inevitable until 2040.

At the same time, the Sydney Morning Herald is noting a zero-emissions plan for Britain – whose parity emissions per capita are minimally four times those of Australia – as Australia approves the polluting Adani coal mine. There is – and was even before last month’s surprise election – a certainty Australia will expand fossil fuels whilst the EU moves to zero-net-emissions.

Many (including my brother) naïvely believe that Australia will eventually be condemned as a pariah state for expanding fossil fuels. Nevertheless, this viewpoint overlooks demographic reality. Australia already has substantially higher total fertility than those nations most advanced in decarbonisation. Recent trends towards lowest-low fertility in Finland (from 1.9 to 1.5 children since 2010) and other European nations whose fertility was the least low during the 2000s suggests that Morrison’s policies will widen this gap.

The fact is that – as I have emphasised for two decades – Australia must ecologically have by far the lowest emissions per capita in the world. This demand places human energy consumption upon its natural biological “footing”. Environment, Capitalism and Socialism demonstrated three decades ago that the money existed to finance a rapid transition to a carbon-free Australia as early as 2005 or 2010 – were major polluters taxed severely enough.

As Dimitri Lafleur has partially shown, a carbon-free Australia would remake the world economy by:

1. radically limiting energy and materials use on a global scale, especially in desert nations with naturally low-energy ecologies and zero hydropower potential
2. shifting “developing” economies towards renewable energy once they do not have cheap fossil fuels from Australia and the oil states
3. shifting energy-intensive industries towards those (Enriched and Tropical) nations with large resources in hydropower
4. shifting agriculture towards the high-latitude nations with youngest and most fertile soils
1. this would occur because land clearing is a major source (around 20 percent) of greenhouse emissions in Australia
2. also, Australian soils are thirty thousand times older and more weathered than soils of most other Quaternary landmasses
3. young, high-latitude areas are also least affected by runaway climate change shown as certain by Nature
5. shifting away from planned obsolescence towards long-lasting consumer goods that use fewer resources over the long term

What needed to be done back in the 1990s was for the globe to recognise that – regardless of its relatively small aggregate emissions that have led even environmentalists to neglect it – a rigid, zero-compromise, zero-emissions target for Australia no later than 2010 would have:

1. largely solved global greenhouse gas emissions by radically altering global development patterns
2. paid for the ecological crisis out of the pockets of those people – alongside the Arab Gulf royal families – with greatest duty and ability to pay
3. achieved this in a manner in agreement with Earth’s natural ecology (smallest per-capita energy consumption and emissions in arid desert nations)
4. in an Enriched World then and now crippled by excessive environmental regulations, which stand likely to achieve negligible global gains while Australia mines and uses more and more coal, created major economic opportunities including:
• phase-out of economically crippling Enriched World farm subsidies as Australia’s unsustainable pastoral and broad-acre farmland would be converted to ecotourism
• revitalising such industries as aluminum and titanium smelting when coal use in phased out in Australia and other nations lacking hydropower potential

Given the experiences of the past quarter-century and especially last month’s election, there exists zero possibility that Australia will ever elect a more environmentally responsive government. Thus, other countries are burdened with the critical task of clamping down on the worst environmental performer – a task entirely ignored but unless achieved even a total carbon phase-out in the EU and East Asia will achieve little in the long term.

Instead of decentralised global environmental protests (as seen in recent weeks) what was needed in 1994 and stands three decades overdue was and is a global focus upon the centre of power in the worst environmental performer: protests demanding uncompromising, rapid decarbonisation of Australia, or a “March on Canberra”. Whilst the effects would not be immediate, would be costly to the rest of the world and would need to be sustained over years and even decades, they possess potential to actually deal with the planet’s worst polluter rather than permit Australia indefinite emissions increases negating large-scale decarbonisation abroad.

A “famous” typo recalled — and Trump beating it

2019 Melbourne rainfall to today. With nothing forecast Melbourne seems very likely to not merely receive less than the city’s existing record low (332.3 millimetres), but receive less than the existing Victorian record low calendar-year total

Victoria’s record low annual rainfall up to 2018 — almost certain to be beaten by a substantial portion of the state in 2019 — from Kyndalyn Park (latitude 34.71° S, longitude 142.93° E)

In recent days with only 43 millimetres of rain in Melbourne for 2019 and a probable record dry year by a factor of four or more — likely Melbourne will beat even Kyndalyn Park’s 1967 state record low rainfall of 75.9 millimetres — I have gleefully recalled a personally famous typo from Perth’s Sunday Times on 27 April, 2008:

What you're paying

Unleaded petrol prices in Brisbane and on the southern Queensland coast today hit $1.46 a litre, according to petrol price monitoring website Motor Mouth.

But unleaded fuel was available at some northern Brisbane outlets for as little as $1.30. Prices on the Gold and Sunshine coasts have also reached $1.46, but some fuel could be found for up to 15 cents a litre less, according to petrol price monitoring website, MotorMouth. Queensland has an eight cents a litre petrol subsidy.

Unleaded prices in Adelaide are nearing $1.55 a litre. Sydney prices range from $1.47 to $1.55 a litre. Melbourne drivers are paying up to $153.90 a litre, while Perth motorists are faring better at $1.40 a litre.”

Seeing this text, nobody would miss the typo. The typist writing for the Sunday Times obviously meant to write

153.90¢ a litre,

which the news.com.au system apparently automatically would change to $1.54 a litre, whilst automatically changing 153.90$ a litre to $153.90 a litre.

Rather than ignoring the typo, I — who has long known Australian petrol as untenably cheap — saw it as one typo I wished were true. Such increased fuel prices would — even with much less fuel sold — give Australia’s governments funding for neglected environmental projects like:

1. complete closure of coal-fired power stations
2. a large-scale renewable energy plan and budget to decarbonise the whole continent
3. a nationwide high-speed rail network to reduce pressure on Tullamarine and Kingsford-Smith Airports
4. public transport networks in all major cities and their surrounding semi-rural districts surpassing the world’s best extant networks
5. complete revegetation of southern Australian farmland certain to be aridified by the rapid poleward widening of the subtropical arid belt

When I told Mummy about the Sunday Times typo a day ago, I was told that Donald Trump has made a much worse blunder when discussing a major Sri Lanka attack on Easter Sunday, whereby eight bombs exploded in churches and hotels. I will quote Trump:

Heartfelt condolences from the people of the United States to the people of Sri Lanka on the horrible terrorist attacks on churches and hotels that have killed at least 138 million people and badly injured 600 more. We stand ready to help!

My mother has always been extreemly critical of President Trump, seeing him as a racist, sexist populist appealing to the emotions of white rural Americans who feel left out with modern technological changes. Mummy, who knew the 2008 typo and my admittedly emotional response to it, said that Trump “surpassed” the Perth newspaper’s typo by a thousand if I recall correctly. Actually, Trump’s error surpassed the 2008 typo by ten thousand. Some of his critics believe Trump magnified the death toll intentionally, presumably to make Americans more scared of Muslims.

However, one difference exists that stands far more critical than the magnitude of the errors. There would exist major benefits to Earth’s ecology, frequently discussed on this blog, from less cheap Australian petrol. There would exist no benefits to anybody or anything from terrorists achieving a death toll corresponding to a nuclear mini-holocaust. If it were true that terrorists could murder on the scale President Trump claimed, there would be the danger of terrorism killing entire civilisations.

Plotting actual versus parity per capita greenhouse gas emissions

In recent posts about the failures of the Kyōtō Protocol, I have expressed extreme anger that:

1. the highest per capita greenhouse gas emitters – excluding a few remote small tropical islands – are desert states whose natural ecologies are based on extreme limitation of energy consumption
2. excepting Least Developed Countries (LDCs) there appears to be an anticorrelation between actual per capita emissions and the relative per capita energy consumption allowed by ecology
3. (1) and (2) are poorly noted by either ecologists or climatologists
• even Tim Flannery, who would be expected to be the most aware person in the world of this, seldom makes the connection so explicit as it should be made

In order to have a reasonable look at the actual situation, I have plotted a graph of actual per capita emissions against those which I believe reflect relative ecological parity of each nation (for methodology see below):

2013 per capita emissions versus relative ecological parity emissions. The actual Pearson correlation coefficient is +0.06

Being aware that very poor Least Developed Countries would distort any observable anticorrelation, I did another experiment without the Least Developed Countries, but Least Developed Countries actually had less effect on the result than I expected. However, when I included only the upper-middle and upper-income countries, I did find a very clear negative correlation between actual and ecological parity per capita emissions:

2013 per capita emissions versus relative ecological parity emissions for upper-middle and upper-income countries. The Pearson correlation coefficient is -0.27

This observed anticorrelation does suggest that in the Enriched World there exist inherent political or resource limitations to the growth of per capita greenhouse emissions that are absent from the arid subtropics of the Eastern Hemisphere or from the humid tropics (e.g. Årts and Janssen, 2006, Cornwallis et. al, 2017). In resource-exporting nations, extremely high energy consumption by the ruling classes is actively supported by suburban citizens who gain a low-tax, high-energy lifestyle based around large, comfortable cars. This is one reason for the exceedingly high emissions of Australia and the Gulf States. In the Enriched World ruling classes lack such cheap energy, while demands by lower classes for redistribution are much stronger. This serves to limit per capita energy consumption to a level frequently exceeded in the resource-richest nations.

Such an anticorrelation is wholly untenable and ignorance thereof a significant problem. It also suggests development in historically poor tropical and desert nations as fraught with severe danger for the planet’s climate.

Methodology for Parity Emissions:

1. Each point of land was aligned to one of the ecoregions outlined in the second Kyōtō Protocol post
1. high mountain regions of the tropics – if wet enough to qualify as humid under the Köppen classification – had the same weighted emissions as the Enriched World
2. in the cases of New Zealand (Milewski, personal communication) and the Iberian Peninsula and southern Mediterranean (McMahon and Finlayson, 1991), ecological peculiarities allowed me to lower the parity emissions from 3 to 2.75
2. For each country, the weighted geometric mean of the parity regions for each point of land was assigned as its parity emissions
3. To make the graph, these parity emissions were graphed on the x-axis against actual emissions from 2013 (World Resources Institute, 2014)

References:

• Cornwallis, Charlie K.; Botero, Carlos A.; Rubenstein, Dustin R.; Downing, Philip A.; West, Stuart A. and Griffin, Ashleigh S.; ‘Cooperation facilitates the colonization of harsh environments’; Nature, Ecology and Evolution, vol. 1 (2017) article 0057
• Flannery, Tim F.; The Future Eaters: An Ecological History of the Australian Lands and People; ISBN 0730104222
• Lovegrove, Barry G.; ‘The Zoogeography of Mammalian Basal Metabolic Rate’; The American Naturalist, vol. 156, no. 2 (August 2000), pp. 201-218
• McMahon, T.A. and Finlayson, B.L.; Global Runoff: Continental Comparisons of Annual Flows and Peak Discharges. ISBN 3-923381-27-1
• World Resources Institute; ‘Climate Analysis Indicators Tool (CAIT) Version 2.0. (Washington, DC: World Resources Institute, 2014)’
• Årts, Paul and Janssen, Dennis; ‘Shades of Opinion: The Oil Exporting Countries and International Climate Politics’; The Review of International Affairs, Vol. 3, No. 2, Winter 2003, pp. 332-351