Transforming the world into a doughnut: a vital clarification

Imagining Toroidal Life as a Sustainable Alternative (Part #6)

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As noted by Les Narvasa (Can We Transform The World Into a Doughnut? Brilliant, 2015): The world can be transformed into a doughnut (sphere becoming a torus, to be more generic) by the mathematics of what topologists now call: surgery theory. This process, described as Ricci flow with surgery, was invented by the mathematician Grigori Perelman -- awarded several distinguished mathematical prizes from 2006 in recognition for this outstanding work (but declined them).

As noted in the introduction, it should be strongly emphasized that this argument is quite distinct from that envisaging the physical Earth as a torus, as criticized by Beckett Mufson (Apparently, Some People Believe the Earth Is Shaped Like a Donut, Motherboard, 13 November 2018) and to a related question and comments (Is the earth a torus? Quora, 2018; Toroidal Earth Society (Reddit, 2016; Anders Sandberg, What would the Earth be like if it was the shape of a donut?, Gizmodo, 2014). A 3D model of that possibility has however been produced (Henry Segerman, Torus Earth: Peirce quincuncial projection, 2016).

Doughnut and boundaries: In a period of global crisis, there is considerable irony to the topological association of the world with a doughnut given the use of the metaphor by Kate Raworth (Doughnut Economics: seven ways to think like a 21st-century economist. 2017; A Safe and Just Space for Humanity: can we live within the doughnut? Oxfam Discussion Papers, 2017; Introducing 'The Doughnut' of social and planetary boundaries for development, Oxfam International, 10 February 2012). This has evoked a preoccupation with doughnut economics in relation to achieving the UN's 17 Sustainable Development Goals. The metaphor featured in a gathering of the World Economic Forum (Kate Raworth, How to do business with doughnuts, 25 January 2018).

The irony is all the greater in that the reference to "planetary boundaries" derives from a presentation to the Club of Rome (Johan Rockström, et al., Planetary Boundaries: exploring the safe operating space for humanity, Ecology and Society, 14, 2009, 2). Curiously the boundaries have been strictly defined in purely physical terms, as challenged in a commentary thereon (Recognizing the Psychosocial Boundaries of Remedial Action: constraints on ensuring a safe operating space for humanity, 2009; Exploring the Hidden Mysteries of Oxfam's Doughnut: recognizing the systemic negligence of an Earth Summit, 2012).

Contrasting the Earth-System boundaries with the boundaries of Remedial Action Capacity
Oxfam Doughnut	Nine planetary boundaries	Nine remedial capacity boundaries
from Kate Raworth, A Safe and Just Space for Humanity: can we live within the doughnut? (2012).	from Rockström, et al. (Planetary Boundaries: exploring the safe operating space for humanity (2009)	from Recognizing the Psychosocial Boundaries of Remedial Action (2009)

The question here is with respect to what understanding of "space-time" and the "planet" any such "boundaries" are to be found. In particular, how is the "safe operating space" then to be understood when the representations are flat, in contrast to any requirement for a global perspective, or one related to time -- and possibly multidimensional in nature, however that is to be understood and comprehended? With respect to a doughnut configuration, how are boundaries to be recognized and comprehended in framing such a space?

Given the toroidal insights explored here, and their association with knots (as discussed below), is there the probability that these boundaries and spaces are highly vulnerable to being "twisted" in topological terms -- at least insofar as many are likely to comprehend them?

Potential confusion: There is every possibility for unfruitful confusion in this complex of associations -- metaphorical and otherwise -- a confusion which may be fundamental to appropriate appreciation of both the challenge and the opportunities. Elements of the confusion include:

• the importance attached by mathematics to the distinction between a 2-sphere and a 3-sphere, where the former defines only the surface of a sphere and the latter includes what is within that surface. As noted by Wikipedia: a 2-sphere is a two-dimensional closed surface embedded in a three-dimensional Euclidean space, whereas a "ball", is a three-dimensional shape that includes the sphere and everything inside the sphere (a closed ball), or, more often, just the points inside, but not on the sphere (an open ball). The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. This is analogous to the situation in the plane, where the terms "circle" and "disk" (of a certain thickness) can also be confounded.
  
  
• whether references to the planet, with mathematical implications, effectively assume that it is a 2-sphere or a 3-sphere (or glome) -- as the first of a series of n-spheres (which may all be termed hyperspheres). The 3-sphere may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogously to how the boundary of a ball in three dimensions is an ordinary sphere (namely a two-dimensional surface, or 2-sphere),, the boundary of a ball in four dimensions is a 3-sphere (an object with three dimensions). A 3-sphere is an example of a 3-manifold and an n-sphere. A 3-sphere is a compact, connected, 3-dimensional manifold without boundary. It is also simply connected. What this means, in the broad sense, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The Poincaré conjecture, proved in 2003 by Grigori Perelman, provides that the 3-sphere is the only three-dimensional manifold (up to homeomorphism) with these properties.
  
  
• framed in this way, the question is under what conditions the planet can be usefully said to have "boundaries" -- and when is this inference dangerously misleading. It is said of the 3-sphere, especially when understood as embedded in 4-dimensional space, that it is finite but unbounded. In the continuing debate about the shape of the universe, one question is whether it should indeed be considered to be finite but unbounded, and closed.
  
  
• the seminal study by Rockstrom, et al (2009) specifically defines boundaries to be human-determined values of the control variable set at a "safe" distance from a dangerous level (for processes without known thresholds at the continental to global scales) or from its global threshold. Determining a safe distance involves normative judgments of how societies choose to deal with risk and uncertainty... The choice of control variable for each planetary boundary was based on our assessment of the variable that on balance may provide the most comprehensive, aggregated, and measurable parameter for individual boundaries. Dos this imply a reference to a planet understood as a 2-sphere or a 3-sphere?
  
  
• there are potential comparisons to be made between human understanding of the shape of the universe and understanding of the shape of the "globe" in framing and containing planetary life as a whole -- especially . given the references above to the "flatness" of the Earth. Cosmologists distinguish between the observable universe and the global universe -- raising the question as to the extent to which planetary life can be understood to be observable, despite the extent of surveillance (and bearing in mind the unexplored ocean depths). Any reference to the global shape of the planet as a system, merits comparison with the distinction made by cosmologists as to whether the global universe is: finite or infinite; flat (no curvature), open (negative curvature), or closed (positive curvature); the degree of connectivity, namely how the universe is put together, i.e., simply connected (like a sphere) or multiply connected (like a torus). A three-torus model of the universe has been proposed (Evelyn Lamb, A Few of My Favorite Spaces: the three-torus, Scientific American, 30 December 2015.
  
  
• even if the focus is on boundaries as understood by the natural sciences, it is clear that the study by Rockström, et al (2009) uses "human-determined values" in determining what amounts to the "radius" of the planet -- although clearly, since nine such boundaries are recognized by the study, this implies a complex situation of co-existing "spherical" boundaries of the planet. How do these relate in mathematical terms to the "globe" -- especially when the challenge of non-physical "boundaries" can be recognized by the "non-natural sciences"? A variant of this question was raised separately with respect to the challenge of the environment and climate change (Are Environmentalists and Climate Scientists in Denial? Climate change recognized as primarily a psychological challenge, 2019)

Extra dimensions? The question raised in this argument, especially given the metaphorical reference to "doughnut" and any implication that "planetary boundaries" are also metaphorical, is whether there is a much greater need for clarity. The associated arguments call for recognition of the dangers of oversimplification in obscuring the extent to which the challenge could be more fruitfully understood as N-dimensional. The "globe" and "planet" in question are then better recognized as being N-spheres, where N may well be as high as the 9 to 11 recognized with respect to the universe (as noted above), That the planet should be understood as having 9 boundaries is perhaps indicative of this -- in one sense at least.

The number of dimensions, the "extra dimensions", may even be more comprehensible with respect to the "planet" as a whole. As understood by physics, through compactification some of the extra dimensions are assumed to "close up" on themselves to form circles. In the limit where these curled up dimensions become "very small", a theory is obtained in which spacetime has effectively a lower number of dimensions.

Is it the case that the natural sciences applied to the challenges of the environment have adopted the convenience of ignoring extra dimensions -- as being so small as to be negligible? Are they however significant to the human experience of the environment, and to the credibility of strategic initiatives? This neglect is strangely mirrored in the comprehensible neglect by most of the many forms of movement in which the planet is understood to be engaged within the universe (Ethan Siegel, Our Motion Through Space Isn't A Vortex, But Something Far More Interesting, Forbes, 30 August 2018; Considering the motion of the Earth, the solar system, and the galaxy, how fast am I moving while lying in bed asleep? Ask an Astronomer).

Requisite flexibility: With respect to the argument here concerning human intuitive understanding of torus versus sphere, the clarification of the above arguments could well enable a fruitful shift in focus from a global to a toroidal perspective -- given the intimate topological relationship between the two (if appropriately understood with mathematical clarification). Can the world indeed be transformed into a doughnut -- as suggested above -- in order to address the complex of issues more effectively, as claimed by "doughnut economics", for example? Will history recognize the current approach to "planetary boundaries" as being tainted by what amounts to "flatterland" thinking?

Of some relevance is the following exchange on a blog maintained by Kate Raworth (Doing the Doughnut at the G20?):

David Needham (1 December 2018): I am fervent about your economic analysis but worry about the name Doughnut – a name with insufficient gravitas to be taken seriously by those who need to take it seriously and have the power to do something? ... In my mind it is Toroid economics.

Laura Kennett (2 December 2018): It's great to see the sharing of ideas to give all of humanity a map towards the doughnut/toroid – I agree the term "doughnut" may lack the credibility and health-consciousness that we need so I like the term "toroid"... Trying to squeeze humanity into a two-dimensional space could be improved by increasing the "volume" or height of the doughnut/toroid. The third dimension could represent the "adaptive capacity" of a nation/state/entity.

Kate Raworth (2 December 2018): Yes, torus is a far less silly name, but almost no one knows what one is!

Whilst "doughnut" may indeed be silly, the issue is whether the manner in which it is so readily known avoids consideration of characteristics of a torus usefully clarified by topology -- notably in the light of the confusion with regard to global, ball and sphere. Use of "doughnut" dangerously inhibits recognition of how these are related and the manner in which one may be transformed into the other -- if Raworth's arguments for "doughnut" are indeed a means of "safeguarding planetary boundaries".

Silly or not, a further concern is whether, such a metaphor is too simple to correspond to the requisite complexity of the situation. As such it lends itself to being characterized as insulting for a civilization dependent on far greater complexity, notably in the light of national enthusiasm for getting into orbit in order to escape the challenges of life on Earth. Given the acknowledged complex of skills required to get into orbit, is there a dangerous naivety in cultivating the illusion that getting strategies to "fly" is as simple as flying to Mars on a doughnut?

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