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Roger Penrose

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Penrose in 2011
The facts of the matter
May the mass times acceleration be with you
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Atoms trying to understand atoms
Science and fun cannot be separated.[1]

Sir Roger Penrose (8 August 1931–) is a British mathematician and a Nobel Prize winning mathematical physicist who was a long-term collaborator of Stephen Hawking. His work is focused on black holes, cosmology and, quite controversially, on quantum consciousness. He has also written several books both for the general public and bible-thick ones for people more familiar with mathematics and physics.[2]

Background and family[edit]

Penrose was born in Colchester, England in 1931. During the World War II he was in Canada. Later he moved back to England, where he attended University College London and graduated with a first class degree in mathematics.[3]

The Penrose family has been well acquainted with science. Penrose's father was a geneticist, his maternal grandfather was a biochemist, his elder brother is a theoretical physicist, his younger brother is a chess Grandmaster who's won the British Chess Championship ten years in a row. His sister is a geneticist and his wife is a mathematician.[3][4]


Penrose did his Master of Sciences and PhD at Cambridge University (both in mathematics). Later he moved to Oxford University, where most of his scientific career has taken place.[5]


Impossible shapes[edit]

Penrose has always liked weird types of geometry, especially the conformal ones which take a huge part in his Conformal Cyclic Cosmology that he devised in 2006. While still a student in the 1950s, he popularised an impossible shape known as 'Penrose triangle'. He later inspired the Dutch artist M. C. Escher to create artworks of other impossible shapes and structures like the WaterfallWikipedia and Ascending and Descending.Wikipedia[6]

Penrose tiling

Penrose tiling[edit]

Penrose managed to create the most optimal solution for a non-periodical tiling. In the 1960s mathematician Hao Wang came up with a solution consisting of 20,426 different tiles. Then he reduced the number of tiles to 104. In 1968 Donald Knuth reduced it to 92 tiles. It was thought that it could not be reduced much more, but then Penrose took a look at it and reduced that number just to two. The pattern can be seen on the pavements in Oxford (outside the Maths Department at the University of Oxford), in Helsinki and on a wide wall of a building in San Francisco.[7][8] The tiling has many properties involving the golden ratio φ (1.618). It also resembles a tiling by Johannes Kepler in many ways.[9]

Dan Shechtman, the chemist who discovered quasi-crystals (the crystals that are arranged in the Penrose tiling way), was awarded the Nobel Prize in Chemistry in 2011.[10]

Penrose diagrams[edit]

Penrose is also known from Penrose diagrams (which he always calls conformal diagrams). A Penrose diagram is an improved versions of classical spacetime diagrams. Their main advantage is that they manage to map all the values (from -infinity to +infinity) to a range between -π and π using trigonometrical functions. Hence, they allow representation of infinite time and space on a finite piece of paper making them very helpful in showing what happens around black holes.[11]

Twistor theory[edit]

Twistor theory is a mathematical theory (though at the intersection of mathematics and physics), constructed by Roger Penrose in 1967, that projects geometric objects from a 4-dimensional spacetime (known as the Minkowski space) into geometric objects in a 4-dimensional complex space with metric (2,2). It's a helpful tool in special theory of relativity.[12]


In 1955 Penrose independently devised a generalisation for inverse matrices.[13] It is widely used in linear algebra to:

  • compute the "best fit" solution to a system of linear equations that lacks a solution.
  • obtain all solutions of a linear system
  • find a minimum norm solution to a linear system
  • define a condition number for any matrix


The first image of a black hole

Penrose began to pursue physics in the 1960s and was a long-time collaborator with another famous scientist, Stephen Hawking, with whom he worked especially on singularities and black holes.

Penrose-Hawking singularity theorems[edit]

In 1963 a paper by Vladimir Belinsky, Isaak Khalatnikov, and Evgeny Lifshitz was released which claimed that no singularities can exist in generic solutions for the Einstein's field equations. The paper was proven false, a year later, by Penrose, who showed that sufficiently dense matter (symmetrical or not) has to collapse into a black hole.[14]

When Georges Lemaître developed his theory which later became known as the Big Bang theory, it had many flaws. The most important one of them was that the universe had to be in a very improbable state in order for it to work. These implausible conditions were later corrected by Hawking using singularity theorems,Wikipedia which he proved together with Penrose.[15]

Penrose and Hawking modified Penrose's original paper on black hole singularities to apply it to the whole universe. These theorems were instrumental in Penrose being awarded the Nobel Prize in Physics in 2020. Had Hawking been alive at the time, it is likely that he would also have received it.[16]

Cosmic censorship hypothesis[edit]

In 1969, Penrose formulated a hypothesis about the universe that stated that one cannot directly see a singularity. In other words, the theorem says that if there is a singularity, it will be always covered by an event horizon.[17] The hypothesis is divided into: weak censorship conjecture and strong censorship conjecture. The strong one was proved false in 2018.[18]

Penrose mechanism[edit]

In 1971 Penrose proposed an idea that energy can be extracted from rotating black holes.[19] This can occur if the rotational energy of the black hole isn't located inside the event horizon. In the same year, the theoretical physicist Yakov Zeldovich translated this idea into mechanical waves. His translation was experimentally confirmed in 2020.[20] Using Penrose mechanism up to 20.07% of the original mass of a black hole can be extracted.

This idea might look like Hawking radiation as they both propose a mechanism for removing mass from black holes. The difference is that Penrose mechanism doesn't work for all black holes (only for the rotating one), while Hawking's one works for every kind of a black hole.

Spin network[edit]

In 1971, Penrose invented the so-called spin network, a type of diagram that can be used to represent states and interactions between particles and fields in quantum mechanics. It is a widely used concept in loop quantum gravityWikipedia (LQG) — String Theory's biggest rival. They are also used in mathematics for several purposes.[21]

Conformal Cyclic Cosmology[edit]

Penrose is widely known for his unconventional approach to cosmology. In 2006, he developed a proposal a cyclic model of the universe, though based on rather unfamiliar things to such models.[22]

Massless ain't fun[edit]

According to general theory of relativity the faster you move, the slower the time passes for that moving object. If two people synchronised their clocks and one of them sat down for a week while the other made a trip around the Moon at half the speed of light, their clocks would not be synchronised when they met again. The clock that saw the other side of the Moon would show that slightly less time had elapsed. The clock of a photon would not tick at all - this is because it moves at the maximum possible speed. In other words: photons experience no time.

The other way to illustrate it is to combine possibly two of the world's most famous equations: E = mc² and E = hν into one: m = hv/c². Since energy is equivalent to mass (E = mc²) and energy is equivalent to frequency (E = hν; frequency can be thought of as a ticking clock), then mass is equivalent to frequency. That means, if one didn't have any mass, one would lose the notion of time. Photons don't have any mass, hence, they feel no time.

What Penrose argues is that in the remote future, the universe will be devoid of mass: everything will be sucked up by black holes. Black holes will evaporate in Hawking radiation (it would take just around googol years) and the universe will consist mainly of photons. Penrose says that after the universe becomes a cold broth of photons, tedium is the most interesting thing about it. But that's actually not so bad because the photons are the only things inside it, and it's quite difficult to bore a photon because it doesn't experience time. But if they don't experience time, neither do they experience distance or scale, the definition of which is based on time. That means there is no difference between a kilometre or a foot. Big becomes equivalent to small, so maybe massless is fun after all.

A light cone

Small bigness, big smallness?[edit]

The other thing that is indisputable among cosmologists is that before the Grand Unification Epoch particles were massless as there was no symmetry breaking then.[23][24]

What that means is that the Big Bang is conformally indistinguishable from the massless remote future of our universe. The word conformally is very important here. You can find a similar concept in geometry. If you draw a triangle on the plane, its angles will always add up to 180 degrees, regardless of its size. In a massless universe, you can imagine a light cone with no particles inside it (everything inside a light cone has mass, the edges of a light cone represent the light rays that don't bear mass). And since the notion of time and distance disappear, you can neglect the metric scale on it. What that causes is that however big the light cone is (with no particles inside it and with no scale), its dimensions become equivalent. An extremely large light cone and an extremely small one make no difference. They are conformally indistinguishable .

Hence, massless Big Bang and massless time-like infinity of our universe are conformally equivalent. You can squash down infinity, because it's conformal, and stretch out the Big Bang and make them indistinguishable. The Big Bang is of course much, much, much hotter, but as you squash down the conformal infinity (which does contain energy, photons for example have energy), you increase the energy density causing it to get hotter and hotter. And conversely, stretching out the Big Bang makes it cooler and cooler.

This idea, so far, probably isn't much opposed, but Penrose takes it one step light year further.

Hello from the other side[edit]

Here's where the actual idea of Conformal Cyclic Cosmology (CCC) starts. According to Penrose, since the infinity of our universe is equivalent to the Big Bang of our universe, the Big Bang was an infinity of a previous universe. Penrose calls the time from the Big Bang to the infinity an aeon. So, according to CCC, the beginning of our aeon was the end of a previous one, and the end of our aeon will be the beginning of another aeon.[25]

The universe (which is what Penrose calls the infinite chain of aeons) in the CCC model doesn't have a beginning. A universe (whatever that means) with no beginning is not what William Lane Craig likes.[citation NOT needed] This caused him to release an entire document in which he claims that CCC is a multiverse model with aeons as 'branches'. Later, he corrected himself, but couldn't help criticising the CCC model again, this time showing that he'd read neither Penrose's paper from 2006, nor watched any lecture of his.[26]

Is there an echo?[edit]

In the first few years after Penrose published his paper in CCC in 2006, he was quite happy that no one could prove him wrong. But around 2010, he came up with an idea to make his idea falsifiable. He thought that colliding supermassive black holes in the previous aeon would release an enormous amount of energy in the form of gravitational waves, which, since they don't have mass, would be visible on the Cosmic Microwave Background Radiation as ring-type structures.[27]

In 2010, Roger Penrose and Vahe Gurzadyan published a paper claiming that observations of the cosmic microwave background made by the Wilkinson Microwave Anisotropy ProbeWikipedia (WMAP) and the BOOMERanG experimentWikipedia contained an excess of concentric circles compared to simulations based on the Λ-driven universe (That is a universe which is exponentially expanding). However, they have used a non-typical methodology which caused a bad reception of the paper.[25]

In 2013 Penrose and Gurzadyan published another paper, related to their previous one, in which the WMAP data was directly analysed. In 2015 they updated it with the newer Planck data which confirmed WMAP results.[28] Later, another paper on ring-type structures in the CMB was published. This time tests were carried out by Daniel An and a group of Polish scientists (Krzysztof Meissner, Pawel Nurowski) in which they claimed that the CMB (from the Planck satellite) shows such structures. They created 1,000 artificial maps and made sure that these maps did not have any previously encoded rings. They then assigned each map a percentage probability of the presence of ring structures. The real CMB showed the highest percentage. They then stretched the rings by turning them into ellipses, and the percentage on the true CMB began to decrease. When the rings were stretched enough, the percentage was about the same on each map.[29]

Later Penrose and Meissner came up with another idea that could falsify the CCC theory, which is based on Hawking radiation. The smaller a black hole is, the faster it evaporates. This means that if the black hole is already really small, it evaporates really fast, making it even smaller and evaporating even faster. At the very end of a black hole's life, the amount of radiation is so great that the black hole explodes in an explosion that Penrose calls a 'pop'. Such 'pops' that occurred in the previous aeon should be (as predicted by the CCC) visible as slightly higher temperature spots on the CMB with an angular diameter of around 4 degrees (or around 8 times the diameter of the Moon). Penrose, An, Meissner and Nurowski published a paper in which they claimed that the CMB (both from Planck and WMAP) shows these spots (called the 'Hawking points' and having the angular diameter of 4 degrees) in the same places. Their work was not published for over a year because they had to carry out other tests and improve on those already done (e.g. increasing the number of artificial CMBs from 1,000 to 10,000). Their work was finally published after the last revision in March 2020.[30] The significance of Hawking points is still being disputed.[31]

Penrose also reflected on Fermi paradox. He stated that if a civilisation was technologically advanced enough to manipulate black holes, it could pass information to the next aeon using them. However, it is not difficult to guess that if there were millions or even thousands of such civilisations, this information could be distorted and overlaid.[32]


Penrose has devised many other mathematical and physical conjectures, theorems, structures and theories. Here's a list of some of them of lesser importance and recognition:



Penrose wasn't brought up in a religious family. He regards himself as agnostic, but when he discussed philosophy and science with William Lane Craig, he called himself an atheist.[34] Despite lacking a belief in God, he believes that the universe is not purposeless.[35]

Cosmic inflation[edit]

Penrose argues against the widely accepted theory of cosmic inflation, claiming that it is flawed. In his book Fashion, Faith and Fantasy,Wikipedia the title 'fantasy' refers to it. Penrose finds inflation problematic when it comes to the uniformity of the universe and the entropy.[36][37] Paul Steinhardt — one of the founding fathers of inflationary cosmology, who later became one of the most recognisable critics of inflation, although not a supporter of CCC wrote: "Penrose considered all the possible configurations of the inflaton and gravitational fields. Some of these configurations lead to inflation … Other configurations lead to a uniform, flat universe directly – without inflation. Obtaining a flat universe is unlikely overall. Penrose's shocking conclusion, though, was that obtaining a flat universe without inflation is much more likely than with inflation – by a factor of 10 to the googol (1010100) power!"[38] Steinhardt also, like Penrose, argues that the Planck satellite data contradicts inflation.[39]

In August 2021 Penrose debated inflation with its founding father Alan Guth.[40]

String theory[edit]

See the main article on this topic: String theory

Penrose prefers loop quantum gravityWikipedia to string theory. One of his mathematical inventions, spin network, is used in LQG. Penrose, and other critics of string theory argue that it is incompatible with Λ-CDM dark energy models (where Λ is cosmological constant introduced by Albert Einstein). Penrose also claims that: "The often frantic competitiveness that this ease of communication engenders leads to band wagon effects, where researchers fear to be left behind if they do not join in."[41] In his book Fashion, Faith and Fantasy, Penrose argues that string theory, and any other theory that assumes more than three spacial dimensions, possess too much functional degree of freedom, making them unbelievably hard to control.[42]


See the main article on this topic: Consciousness

Penrose postulates that consciousness originates at the quantum level inside neurons, rather than the conventional view that it is a product of connections between neurons. Penrose's quantum consciousness hypothesis is certainly built upon a much better understanding of quantum mechanics than it is in other cases (but this does not, of course, make the model true). He argues that the very existence of Gödel's incompleteness theorems demonstrated that the mind had the capability of thinking outside of an algorithmic fashion, i.e. that consciousness is non-computable.[43] Penrose together with an anaesthesiologist Stuart Hameroff developed an idea known as Orchestrated Objective Reduction (Orch-OR). However, anaesthesia's action differs from the model in Penrose and Hameroff's hypothesis, casting serious doubt on the idea.[44]


  1. Penrose's approach to science
  2. Roger Penrose, The Road to Reality, 2004 ISBN 0224044478
  3. 3.0 3.1 Penrose's biography, Maths history, 7 October 2020
  4. Penrose's family part 2, Abingdon School 6 July 2012.
  5. "Oxford Mathematician Roger Penrose jointly wins the Nobel Prize in Physics | University of Oxford Retrieved 3 August 2021.
  6. Ascending and Descending by M.C. Escher – Facts about the Painting. Totally History. 21 May 2013
  7. See the Wikipedia article on Penrose tiling.
  8. Penrose Tilling on a building wall, Photo: Pinterest
  9. Tilings and Patterns, Grünbaum, Branko; Shephard ISBN 9780716711933
  10. Dan Shechtman Nobel Prize, Wikipedia, Retrieved 3 August 2021
  11. Penrose, Roger (15 January 1963). "Asymptotic properties of fields and space-times". Physical Review Letters. 10 (2) doi:10.1103/PhysRevLett.10.66.
  12. Penrose, R. (1967). "Twistor Algebra". Journal of Mathematical Physics. 8 (2): 345–366.
  13. "A generalized inverse for matrices", Roger Penrose, Cambridge 1955
  14. Scientifc Background on the Nobel Prize in Physics 2020, October 2020
  15. Starobinsky, Alexei A. (1980). "A new type of isotropic cosmological models without singularity". Physics Letters B. 91 (1): 99–102.
  16. The Nobel Prize in Physics 2020, 6 October 2020
  17. Cosmic censorship conjecture
  18. "Mathematicians Disprove Conjecture Made to Save Black Holes". Quanta Magazine, Hartnett, Kevin (17 May 2018).
  19. R. Penrose and R. M. Floyd, "Extraction of Rotational Energy from a Black Hole", Nature Physical Science 229, 177 (1971)
  20. Physicists Verify Half-Century-Old Theory about Rotating Black Holes, 24 June 2020
  21. R. Penrose (1971a), "Angular momentum: an approach to combinatorial spacetime," in T. Bastin (ed.), Quantum Theory and Beyond, Cambridge University Press, 1971
  22. "Before the Big Bang: An Outrageous New Perspective and its Implications for Particle Physics", Roger Penrose 2006, Proceedings of the EPAC 2006, Edinburgh, Scotland: 2759–2762.
  23. The Big Bang Started by the Creation of the Reverse Higgs Field, Ding-Yu Chung, Journal of Modern Physics 6(9):1189-1194, August 2015
  24. Higgs boson and the history of the universe,
  25. 25.0 25.1 Gurzadyan VG; Penrose R. "Concentric circles in WMAP data may provide evidence of violent pre-Big-Bang activity". 16 November 2010 arXiv:1011.3706
  26. Roger Penrose and Cosmology, Reasonable Faith 18 December 2016
  27. Roger Penrose, Cycles of Time: An Extraordinary New View of the Universe, 2010 ISBN 9780224080361
  28. Gurzadyan, VG; Penrose, R (2013). "On CCC-predicted concentric low-variance circles in the CMB sky". Eur. Phys. J. Plus. 128 (2): 22. arXiv:1302.5162
  29. Ring Type Structures in the Planck map of the CMB, Daniel An, Krzysztof A. Meissner, Pawel Nurowski Reaserchgate, July 2013. DOI:10.1093/mnras/stx2299
  30. Apparent evidence for Hawking points in the CMB Sky Daniel An, Krzysztof A. Meissner, Pawel Nurowski and Roger Penrose, Monthly Notices of the Royal Astronomical Society, 495 (3): 3403–3408. 2018 (2020) arXiv:1808.01740doi:10.1093/mnras/staa1343
  31. "Re-evaluating evidence for Hawking points in the CMB" Jow, Dylan L.; Scott, Douglas (2020-03-09). Journal of Cosmology and Astroparticle Physics. 2020 (3): 021. doi:10.1088/1475-7516/2020/03/021arXiv:1909.09672
  32. Gurzadyan, V.G.; Penrose, R. (2016). "CCC and the Fermi paradox". Eur. Phys. J. Plus. 131: 11. arXiv:1512.00554Bibcode:2016EPJP..131...11Gdoi:10.1140/epjp/i2016-16011-1S2CID 73537479.
  33. Proof of the Riemannian Penrose Inequality Using the Positive Mass Theorem, Hubert L. Bray, Project euclid, October 2001
  34. Penrose and Craig in a discussion, Unbelievable?, 4 October 2019
  35. A brief history of time (1991) Archived 24 September 2015
  36. Penrose, Roger (2004). The Road to Reality: A Complete Guide to the Laws of the Universe. London: Vintage Books, p. 755. ISBN 0224044478
  37. Penrose, Roger (1989). "Difficulties with Inflationary Cosmology". Annals of the New York Academy of Sciences. 271: 249–264 Bibcode:1989NYASA.571..249Pdoi:10.1111/j.1749-6632.1989.tb50513.xS2CID 122383812
  38. Steinhardt, Paul J. (2011). "The inflation debate: Is the theory at the heart of modern cosmology deeply flawed?". Scientific American. 304 (4): 18–25. Bibcode:2011SciAm.304d..36Sdoi:10.1038/scientificamerican0411-36PMID 21495480
  39. Ijjas, Anna; Steinhardt, Paul J.; Loeb, Abraham (2013). "Inflationary paradigm in trouble after Planck2013". Physics Letters B. 723 (4–5): 261–266 arXiv:1304.2785Bibcode:2013PhLB..723..261Idoi:10.1016/j.physletb.2013.05.023S2CID 14875751
  40. Multiverse or Cyclic Universe ? Alan Guth vs Roger Penrose, skydivephil, YouTube, 16 August 2021
  41. Penrose, The Road To Reality, 2004. p. 1018
  42. How to deal with Roger Penrose's functional degree of freedom objection against higher dimensional theories?, Physics overflow, 24 December 2016
  43. Roger Penrose The Emperor's New Mind (1989) Oxford University Press ISBN 9780198519737
  44. Scientists Discovered Where Anesthesia Works On the Brain, The Science Times, Sandra Handy, 13 April 2019