A mass is a clock: E = mc² and E = hf together imply that mass and frequency are equivalent, so any massive particle oscillates at a frequency determined by its mass and serves as a natural clock.

Multiple black holes are not the same singularity; different stellar collapses produce different, distinct singularities.

The paradoxical geometric figures (tri-bar, impossible staircase) have a limited explanatory relationship to the tiling problem; they represent different mathematical phenomena (visual representation of paradox vs. uncoverable plane-tiling configurations).

A two-state quantum system (e.g., a particle in superposition of position A or position B) will, according to the standard Copenhagen interpretation, exist in a superposition state where it is simultaneously partly in each position until measurement; this can be visualized using Dirac's chalk analogy of a particle being partly here and partly there.

If you lay down rules of proof that could be checked by a computer, you can construct a Gödel sentence which, if you trust the rules, you can see is true by virtue of your belief that the rules only give truths, yet that statement is underivable using the rules themselves.

Understanding, as distinct from mere rule-following, is not a computational process because it involves grasping the truth of statements that transcend any finite formal system.

Polyomino tiling problems demonstrate non-computability: given a finite set of shapes, the question 'Can these shapes tile the infinite plane?' has a definite yes-or-no answer, but there is no algorithm that can compute it—the answer is true or false, but not algorithmically determinable.

Penrose designed a set of Penrose tiles (two shapes) that tile the plane non-periodically—they can cover the plane in a non-repeating pattern that extends indefinitely while following strict adjacency rules, unlike periodic tilings where the pattern repeats.

Mathematical reality exists independently in a Platonic realm: mathematical truths are discovered rather than invented, they exist prior to and independent of human minds, and their truths are there to be found if we are lucky.

Only a tiny part of the mathematical world has direct relevance to the physical world; the majority of mathematics is abstract play, yet the physical world operates according to the specific small subset of mathematics that does apply to it.

Consciousness appears in a tiny part of the physical world; most matter (rocks, etc.) shows no consciousness, while certain entities (humans and some animals) possess consciousness, yet consciousness itself is distinct from the physical laws that govern the rest of nature.

Quantum mechanics is formally an inconsistent theory: the Schrödinger equation describes deterministic evolution of the quantum state, but measurement produces non-deterministic probabilistic outcomes following rules outside the equation, creating an unresolved inconsistency.

The Wigner interpretation (consciousness causes wave function collapse) has been held by Wigner and others, though Wigner was not dogmatic about it; however, Penrose's view is nearly opposite: an objective physical process involving gravity causes collapse, not conscious observation.

When a quantum system reaches a certain lifetime (determined by a mathematical formula involving mass and gravitational effects), it will undergo spontaneous collapse into one or another definite state with certain probabilities, without requiring external measurement or conscious observation.

From a photon's perspective, traveling at light speed, there is no passage of time due to relativistic time dilation at light speed; distance and duration become zero for the photon.

In special and general relativity, there is no universal concept of 'now' or simultaneous events across different reference frames; whether two space-like separated events are simultaneous depends on the observer's frame of reference.

An outside observer watching someone descend into a black hole would see them slow, fade, and asymptotically approach the event horizon, never crossing it from the external perspective; yet the person falling would not notice anything special at the horizon and would cross it in finite proper time.

Penrose's cyclical cosmology model (conformal cyclic cosmology, or CCC) proposes that the big bang was not the beginning but part of an endless sequence of aeons; the model involves exponential expansion driven by dark energy and subsequent reconception as a new big bang.

Einstein introduced the cosmological constant in 1917 to keep the universe static (for philosophical preference), but later called it his biggest blunder when convinced of universal expansion; however, the cosmological constant now appears to be real, driving exponential acceleration.

In the very remote future of the universe, black holes will have evaporated via Hawking evaporation, and the universe will be dominated by photons and electromagnetic radiation, with effectively zero mass.