Peano arithmetic

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# Contents #
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## Idea

Peano arithmetic refers to a [[theory]] which formalizes [[arithmetic]] operations on the [[natural numbers]] $\mathbb{N}$ and their properties. There is a [[first-order theory|first-order]] Peano arithmetic and a [[second-order logic|second-order]] Peano arithmetic, and one may speak of Peano arithmetic in higher-order type theory.

As first-order logic has certain [[syntax|syntactic]] and [[model theory|model-theoretic]] advantages over second-order logic, and has been much further developed, the default notion of Peano arithmetic is usually taken to be the first-order one. However, we will describe both the first- and second-order notions. Note that Peano's original treatment was second-order.

## Second-order Peano arithmetic

Against a fixed background of "[[sets]]" (which we consider [[category theory|categorically]], e.g., a model of [[ETCS]], or even just any [[topos]]), Peano arithmetic is formalized by stipulating a set $N$ with an element $0: 1 \to N$ and an unary [[function]] $s: N \to N$, subject to the following [[axioms]]:

* $\forall_x \neg s(x) = 0$;

* $\forall_{x, y} s(x) = s(y) \Rightarrow x = y$;

* (Second-order [[induction]]) $\forall_{I \subseteq N} [0 \in I \wedge (\forall_x x \in I \Rightarrow s(x) \in I] \Rightarrow I = N$.

Regarding $N$ as an [[algebra of an endofunctor]] $F(X) = 1 + X$, the first two axioms say that the structure map $(0, s): 1 + N \to N$ is [[monomorphism|monic]], and the [[induction]] axiom says that the only $F$-subalgebra of $N$ is $N$ itself. (In particular, $(0, s): F N \to N$ is tautologically an $F$-[[algebra of an endofunctor|algebra]] map where $F N$ is given the $F$-algebra structure $F(0, s): F F N \to F N$. Being [[monomorphism|monic]], the subalgebra $(0, s): F N \to N$ is an [[isomorphism]], by induction.)

It may be shown that these properties determine $N$ up to isomorphism (one says second-order Peano arithmetic is 'categorical', but this is not at all related to the notion of [[category]]), and moreover:

+-- {: .num_prop}
###### Proposition
In a [[topos]], an object which satisfies the Peano axioms is a [[natural numbers object]] (and natural numbers objects satisfy the Peano axioms).
=--

This is a special case of a much more general result which is part of the general theory of [[recursion]] and [[well-founded coalgebras]]:

+-- {: .num_theorem}
###### Theorem
If $\mathbf{E}$ is a topos and $F: \mathbf{E} \to \mathbf{E}$ is a [[taut functor]], then any $F$-[[algebra of an endofunctor|algebra]] $X$ whose structure map $\xi: F X \to X$ is [[monomorphism|monic]] and for which the only $F$-subalgebra of $X$ is $X$ itself is an [[initial object|initial]] $F$-algebra.
=--

## First-order Peano arithmetic

Here we cannot [[quantifier|quantify]] over [[subsets]], as in the second-order induction scheme, and the initial algebra techniques that can be used to manufacture addition and multiplication do not apply. Instead, addition and multiplication need to be built into the [[signature]]. Induction becomes an induction scheme over formulas in the language generated by the signature.

Thus, the signature of first-order Peano arithmetic consists of a constant $0$, an unary function symbol $s$, and binary function symbols $+, \cdot$. The Peano axioms are

1. $\forall_x \neg s(x) = 0$;

1. $\forall_{x, y} s(x) = s(y) \Rightarrow x = y$;

1. $\forall_x \left( x = 0 \vee \exists_y x = s(y) \right)$;

1. $\forall_x x + 0 = x$,

1. $\forall_{x, y} x + s(y) = s(x+y)$;

1. $\forall_x x \cdot 0 = 0$;

1. $\forall_{x, y} x \cdot s(y) = x \cdot y + x$;

1. (Induction scheme) For all formulas $\phi(x, y_1, \ldots, y_k) = \phi(x; \vec{y})$ with $k+1$ free variables,
$$\forall_{y_1, \ldots, y_k} [\phi(0, \vec{y}) \wedge \forall_x (\phi(x; \vec{y}) \Rightarrow \phi(s(x); \vec{y})] \Rightarrow \forall_z \phi(z, \vec{y}).$$

The third axiom is actually redundant, being an instance of the induction scheme (take $\phi(x) = [x = 0 \vee \exists_y x = s(y)]$). We include it because it is needed for weaker systems of arithmetic in which the induction schema is curtailed or dropped, notably [[Robinson arithmetic]].

In addition, the additional parameters $\vec{y}$ are not needed in the induction scheme; adding the induction scheme for first-order formulae $\phi(x)$ where the only parameter is $x$:

* (Induction scheme) For all formulas $\phi(x)$,
$$[\phi(0) \wedge \forall_x (\phi(x) \Rightarrow \phi(s(x))] \Rightarrow \forall_z \phi(z).$$

results in an equivalent theory to Peano arithmetic. See [Parameters in arithmetic induction axiom schemas](https://mathoverflow.net/questions/79174/parameters-in-arithmetic-induction-axiom-schemas).

By the [[Lowenheim-Skolem theorem]], there are [[nonstandard model of arithmetic|"nonstandard" models]] of any infinite cardinality of Peano arithmetic, all of them [[elementary embedding|elementarily embedding]] the standard model of arithmetic.

### Formalization

We present here a formalization of first-order Peano arithmetic in a [[deduction system]] using [[inference rules]]. Such a formalization is important because first-order Peano arithmetic is needed to formally define the universe levels of the [[universe hierarchy]] of [[Russell universes]] or [[Coquand universes]] in many [[dependent type theories]], such as the dependent type theory in the [[HoTT book]].

First-order Peano arithmetic, as a [[first-order theory]] with [[sorts]], consists of the following [[judgments]]

* $\Gamma \; \mathrm{ctx}$, that $\Gamma$ is a [[context]]

* $\phi \; \mathrm{prop}$, that $\phi$ is a [[proposition]],

* $\phi \; \mathrm{true}$, that $\phi$ is a [[true]] proposition,

* $A \; \mathrm{sort}$, that $A$ is a [[sort]],

* $a \in A$, that $a$ is in sort $A$.

In addition, we have rules stating that one could form the empty context, and one could add true proposition judgments and element judgments to the list of contexts:

$$\frac{}{() \; \mathrm{ctx}} \qquad \frac{\Gamma \; \mathrm{ctx} \quad \Gamma \vdash P \; \mathrm{prop}}{(\Gamma, P \; \mathrm{true}) \; \mathrm{ctx}} \qquad \frac{\Gamma \; \mathrm{ctx} \quad \Gamma \vdash A \; \mathrm{sort}}{(\Gamma, a \in A) \; \mathrm{ctx}}$$

We also have rules saying that given a context with a true proposition or element judgment, said judgment holds:

$$\frac{(\Gamma, P \; \mathrm{true}, \Delta) \; \mathrm{ctx}}{(\Gamma, P \; \mathrm{true}, \Delta) \vdash P \; \mathrm{true}} \qquad \frac{(\Gamma, x \in A, \Delta) \; \mathrm{ctx}}{(\Gamma, x \in A, \Delta) \vdash x \in A}$$

In addition, we have the [[natural deduction]] [[inference rules]] for [[classical logic|classical]] [[propositional logic]]:

* Inference rules for [[true]]:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \top \; \mathrm{prop}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \top \; \mathrm{true}}$$

* Inference rules for [[false]]:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \bot \; \mathrm{prop}} \qquad \frac{\Gamma \vdash P \; \mathrm{prop}}{\Gamma, \bot \; \mathrm{true} \vdash P \; \mathrm{true}}$$

* Inference rules for [[logical conjunction]]:

$$\frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma \vdash P \wedge Q \; \mathrm{prop}} \qquad \frac{\Gamma \vdash P \; \mathrm{true} \quad \Gamma \vdash Q \; \mathrm{true}}{\Gamma \vdash P \wedge Q \; \mathrm{true}} \qquad \frac{\Gamma \vdash P \wedge Q \; \mathrm{true}}{\Gamma \vdash P \; \mathrm{true}} \qquad \frac{\Gamma \vdash P \wedge Q \; \mathrm{true}}{\Gamma \vdash Q \; \mathrm{true}}$$

* Inference rules for [[disjunction]]:

$$\frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma \vdash P \vee Q \; \mathrm{prop}} \qquad \frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma, P \; \mathrm{true} \vdash P \vee Q \; \mathrm{true}} \qquad \frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma, Q \; \mathrm{true} \vdash P \vee Q \; \mathrm{true}}$$

$$\frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop} \quad \Gamma, P \vee Q \; \mathrm{true} \vdash R \; \mathrm{prop} \quad \Gamma, P \; \mathrm{true} \vdash R \; \mathrm{true} \quad \Gamma, Q \; \mathrm{true} \vdash R \; \mathrm{true}}{\Gamma, P \vee Q \; \mathrm{true} \vdash R \; \mathrm{true}}$$

* Inference rules for [[implication]]:

$$\frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma \vdash P \Rightarrow Q \; \mathrm{prop}} \qquad \frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma, P \; \mathrm{true} \vdash Q \; \mathrm{true}}{\Gamma \vdash P \Rightarrow Q \; \mathrm{true}} \qquad \frac{\Gamma \vdash P \Rightarrow Q \; \mathrm{true}}{\Gamma, P \; \mathrm{true} \vdash Q \; \mathrm{true}}$$

* Inference rules for [[excluded middle]]

$$\frac{\Gamma \vdash P \; \mathrm{prop}}{\Gamma \vdash P \vee (P \Rightarrow \bot) \; \mathrm{true}}$$

First-order Peano arithmetic consists of one sort $\mathcal{N}$:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathcal{N} \; \mathrm{sort}}$$

for which it is possible to quantify over elements of $\mathcal{N}$:

* Inference rules for the [[universal quantifier]] on $\mathcal{N}$:

$$\frac{\Gamma \; \mathrm{ctx} \quad \Gamma, x \in \mathcal{N} \vdash P(x) \; \mathrm{prop}}{\Gamma \vdash \forall x \in \mathcal{N}.P(x) \; \mathrm{prop}} \qquad \frac{\Gamma \; \mathrm{ctx} \quad \Gamma, x \in \mathcal{N} \vdash P(x) \; \mathrm{true}}{\Gamma \vdash \forall x \in \mathcal{N}.P(x) \; \mathrm{true}} \qquad \frac{\Gamma \; \mathrm{ctx} \quad \Gamma \vdash \forall x \in \mathcal{N}.P(x) \; \mathrm{true}}{\Gamma, x \in \mathcal{N} \vdash P(x) \; \mathrm{true}}$$

* Inference rules for the [[existential quantifier]] on $\mathcal{N}$:

$$\frac{\Gamma \; \mathrm{ctx} \quad \Gamma, x \in \mathcal{N} \vdash P(x) \; \mathrm{prop}}{\Gamma \vdash \exists x \in \mathcal{N}.P(x) \; \mathrm{prop}}$$

$$\frac{\Gamma \; \mathrm{ctx} \quad \Gamma, x \in \mathcal{N} \vdash P(x) \; \mathrm{prop} \quad \Gamma \vdash x \in \mathcal{N} \quad \Gamma \vdash P(x) \; \mathrm{true}}{\Gamma \vdash \exists x \in \mathcal{N}.P(x) \; \mathrm{true}}$$

$$\frac{\Gamma \; \mathrm{ctx} \quad \Gamma, x \in \mathcal{N} \vdash P(x) \; \mathrm{prop} \quad \Gamma, \exists x \in \mathcal{N}.P(x) \; \mathrm{true} \vdash Q \; \mathrm{prop} \quad \Gamma, x \in \mathcal{N}, P(x) \; \mathrm{true} \vdash Q \; \mathrm{true}}{\Gamma, \exists x \in \mathcal{N}.P(x) \; \mathrm{true} \vdash Q \; \mathrm{true}}$$

as well as compare elements of $\mathcal{N}$ for [[propositional equality]]:

* Inference rules for propositional equality on $\mathcal{N}$:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x \in \mathcal{N}, y \in \mathcal{N} \vdash x = y \; \mathrm{prop}} \quad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \forall x \in \mathcal{N}.x = x \; \mathrm{true}}$$

$$\frac{\Gamma \; \mathrm{ctx} \quad \Gamma, x \in \mathcal{N}, y \in \mathcal{N} \vdash P(x, y) \; \mathrm{prop}}{\Gamma \vdash \left(\forall x \in \mathcal{N}.P(x, x)\right) \Rightarrow \left(\forall x \in \mathcal{N}.\forall y \in \mathcal{N}.x = y \Rightarrow P(x, y)\right) \; \mathrm{true}}$$

Then we have the [[axioms]] and [[signature (in logic)|signatures]] for Peano arithmetic, which are given by the following inference rules:

* Inference rules for successor:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x \in \mathcal{N} \vdash s(x) \in \mathcal{N}}$$
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \forall x \in \mathcal{N}.\neg (s(x) = 0) \; \mathrm{true}}$$
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \forall x \in \mathcal{N}.\forall y \in \mathcal{N}.(s(x) = s(y)) \Rightarrow (x = y) \; \mathrm{true}}$$

* Inference rules for addition:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x \in \mathcal{N}, y \in \mathcal{N} \vdash x + y \in \mathcal{N}}$$
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \forall x \in \mathcal{N}.x + 0 = x \; \mathrm{true}}$$
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \forall x \in \mathcal{N}.\forall y \in \mathcal{N}.x + s(y) = s(x + y) \; \mathrm{true}}$$

* Inference rules for multiplication:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, x \in \mathcal{N}, y \in \mathcal{N} \vdash x \cdot y \in \mathcal{N}}$$
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \forall x \in \mathcal{N}.x \cdot 0 = 0 \; \mathrm{true}}$$
$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \forall x \in \mathcal{N}.\forall y \in \mathcal{N}.x \cdot s(y) = x \cdot y + x \; \mathrm{true}}$$

* Axiom schema of induction:

$$\frac{\Gamma \; \mathrm{ctx} \quad \Gamma, x \in \mathcal{N} \vdash P(x) \; \mathrm{prop}}{\Gamma \vdash \left(P(0) \wedge \forall x \in \mathcal{N}.P(x) \Rightarrow P(s(x))\right)\Rightarrow \forall x \in \mathcal{N}.P(x) \; \mathrm{true}}$$

This concludes the formalization of first-order Peano arithmetic.

## Related concepts

* [[Robinson arithmetic]]

* [[Heyting arithmetic]]

* [[nonstandard model of arithmetic]]

* [[Nelson arithmetic]]

## References

Named after [[Giuseppe Peano]]; originating in

* [[Richard Dedekind]], *Was sind und was sollen die Zahlen?* (1888) [scan: [pdf](https://rcin.org.pl/Content/37485/WA35_52499_2264brosz_Was-sind.pdf), [doi:10.1007/978-3-663-02788-1](https://doi.org/10.1007/978-3-663-02788-1)]

* [[Richard Dedekind]] (transl. by W. Beman), _The nature and meaning of numbers_, Chapter II in: *Essays on the Theory of Numbers*, Chicago (1901) [[Project Gutenberg](http://www.gutenberg.org/ebooks/21016), [pdf](https://www.gutenberg.org/files/21016/21016-pdf.pdf)]

* [[Giuseppe Peano]], *Arithmetices principia, nova methodo exposita*, [[Wikipedia](https://en.wikipedia.org/wiki/Arithmetices_principia,_nova_methodo_exposita)]

Review:

* [[David E. Joyce]], *The Dedekind/Peano Axioms* (2005) [[pdf](http://aleph0.clarku.edu/~djoyce/numbers/peano.pdf)]

* Wikipedia, _[Peano axioms](http://en.wikipedia.org/wiki/Peano_axioms)_

[[!redirects Peano axioms]]

[[!redirects Dedekind-Peano axioms]]


Peano curve

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## Idea

The original "Peano (space-filling) curve" is a _[[surjective]]_ [[continuous function]] $I \to I^2$, from the [[closed interval]] $I = [0, 1]$ to the [[product space|product]] with itself, the [[square]]. The existence of such an entity (due to Peano) came as a surprise.

One may characterize exactly which [[Hausdorff spaces]] arise as the continuous [[images]] of a unit interval. These are called *Peano spaces*.

One can similarly show that there is a continuous surjection $\mathbb{R}^1 \to \mathbb{R}^2$ from the [[real line]] to the [[plane]] (both regarded as [[Euclidean spaces]] equipped with their [[metric topology]]), and similarly characterize which spaces arise as continuous images of the real line. These are sometimes called *$\sigma$-Peano spaces*.

Notice that, while of course there is also an injection $\mathbb{R} \to \mathbb{R}^2$, there is _no_ [[homeomorphism]] between these two spaces, or generally between [[Euclidean spaces]] of differing [[dimension]]. This is the statement of _[[topological invariance of dimension]]_.

## Construction

There are many constructions of space-filling curves, but one of the quickest is due to Lebesgue and is closely connected with the Cantor-Lebesgue function.

[[Cantor space]] (following the "middle thirds" construction) can be described as the subspace $C$ of $[0, 1]$ consisting of points whose base-$3$ representation $.a_1 a_2 a_3 \ldots = \sum_{i=1}^{\infty} \frac{a_i}{3^i}$ has $a_i \in \{0, 2\}$ for all $i$ (no $1$\'s). Define a function $\phi: C \to I$ by

$$\phi\left(\sum_{i=1}^\infty \frac{a_i}{3^i}\right) = \sum_{i=1}^\infty \frac{a_i/2}{2^i}$$

(i.e., replace $2$'s in the base $3$ representation by $1$'s and reinterpret the sequence as representing a number in base $2$). Notice that $\phi$ maps the two endpoints of any one of the open intervals removed during the middle-thirds construction to the same point, e.g., for the first middle third $(\frac1{3}, \frac{2}{3})$ we have

$$\phi(.02222222\ldots_3) = .01111111\ldots_2 = 10000000\ldots_2 = \phi(.20000000\ldots_3).$$

In any case, it is very easy to see that $\phi: C \to I$ is a continuous surjective map.

Now: $C$ is homeomorphic to the [[product space]] $2^\mathbb{N}$, a countable product of copies of the [[discrete space]] $2 = \{0, 1\}$. Of course we also have a bijection $\mathbb{N} \cong \mathbb{N} + \mathbb{N}$, inducing a homeomorphism

$$2^\mathbb{N} \cong 2^{\mathbb{N} + \mathbb{N}} \cong 2^\mathbb{N} \times 2^\mathbb{N}$$

and hence a "pairing function" $pair: C \cong C \times C$ that is a homeomorphism (see [[Jonsson-Tarski algebra]]). We use this to construct a continuous surjection

$$C \stackrel{pair}{\to} C \times C \stackrel{\phi \times \phi}{\to} I \times I,$$

denoted say $g: C \to I \times I$, and Lebesgue's idea is to extend $g$ to a continuous function $f: I \to I \times I$ by linear interpolation: if $x \in I$ belongs to one of the open intervals $(a, b)$ removed during the middle thirds construction, say $x = t a + (1 - t)b$ for some $t \in (0, 1)$, then define

$$f(x) = t g(a) + (1 - t)g(b).$$

+-- {: .num_prop}
###### Proposition
The function $f: I \to I \times I$ thus defined is surjective and continuous.
=--

+-- {: .proof}
###### Proof
Surjectivity follows from the fact that its restriction $g: C \to I \times I$ is surjective.

Obviously for each open interval removed in the middle thirds construction, $f$ is continuous at each [[interior point]] $x$ (being locally an [[affine map]] there), and so it remains to check that $f$ is continuous at each point of $C$. So let $a \in C$, and let us prove that $f(x)$ approaches $f(a)$ as $x$ approaches $a$ from the right; a similar argument will prove continuity from the left. This is obvious if $a$ is the left endpoint of one of the removed open intervals, again because $f$ is affine to the immediate right of $a$. If not, then $a$ is a limit from the right of points of $C$. Now $g: C \to I \times I$ is continuous, so given $\epsilon \gt 0$ there exists $\delta \gt 0$ such that ${|g(x) - g(a)|} \lt \epsilon$ whenever $x \in [a, a + \delta)$ and $x \in C$. What if $x \notin C$? Shrink $\delta$ a little more, and assume $a + \delta$ is a right-hand endpoint of a removed open interval, and consider the case where $x \in [a, a + \delta)$ and $x \notin C$, say $x \in (b, c)$ where $(b, c)$ is a removed open interval and $b, c \in (a, a + \delta]$. Then we get the same $\epsilon$-bound as before: putting $x = t b + (1 - t)c$, we have

$$\array{
{|f(x) - f(a)|} & = & {\left|[t g(b) + (1 - t)g(c)] - g(a)\right|} \\
 & = & {\left|t(g(b) - g(a)) + (1 - t)(g(c) - g(a))\right|} \\
 & \leq & t{\left|g(b) - g(a)\right|} + (1 - t){\left|g(c) - g(a)\right|} \\
 & \lt & t\epsilon + (1 - t)\epsilon = \epsilon
}$$

which completes the demonstration.
=--

The same method can be used to exhibit a space-filling curve $I \to I^S$ for any set $S$ of finite or countable [[cardinality]]. Note that in the case where $S$ is a singleton, where we extend the surjection $C \to I$ to $I \to I$ by linear interpolation, we get the [Cantor-Lebesgue function](https://en.wikipedia.org/wiki/Cantor_function).

## Hahn-Mazurkiewicz theorem

The eponymous theorem may be stated as follows:

+-- {: .num_theorem}
###### Theorem
A [[Hausdorff space]] $X$ admits a [[continuous function|continuous]] [[surjection]] $f: I \to X$ from the [[closed interval]] $I$ if and only if it is a [[connected space|connected, locally connected]] [[compact space|compact]] [[metrizable space]].
=--

(N.B. According to the nLab, connected spaces are [[inhabited set|nonempty]]!)

The "only if" half is relatively easy; see [here](https://ncatlab.org/nlab/show/connected+space#arcconnectedness) for some details. The "if" half is rather more involved, but [Willard's General Topology](https://www.amazon.com/dp/0486434796/?tag=stackoverfl08-20) contains a proof. A space $X$ satisfying the stated conditions is called a **Peano space**.

Given this characterization, it is not difficult to characterize which spaces are continuous images of $\mathbb{R}$:

+-- {: .num_theorem}
###### Theorem
A [[path-connected space|path-connected]] Hausdorff space $X$ admits a continuous surjection $\mathbb{R} \to X$ if and only if it is a $\sigma$-Peano space, i.e., a countable union $\bigcup_{n: \mathbb{N}} A_n$ of Peano spaces.
=--

+-- {: .proof}
###### Proof
The "only if" half being fairly obvious, the "if" part may be proved as follows. Since there are continuous surjections $[0, \infty) \to \mathbb{R}$ and $\mathbb{R} \to [0, \infty)$, it suffices to show that a $\sigma$-Peano space admits a continuous surjection from $[0, \infty)$. For each $n \in \mathbb{N}$ choose a continuous surjection $f_n: [2 n, 2 n + 1] \to A_n$. Then for each $n$ choose a path $g_n: [2 n + 1, 2 n + 2] \to X$ such that $g_n(2 n + 1) = f_n(2 n + 1)$ and $g_n(2 n + 2) = f_{n+1}(2 n + 2)$. Then the $f_n$ and $g_n$ paste together to form a continuous surjection $[0, \infty) \to X$.
=--

An example of such a space is the [[Warsaw circle]].

## Related concepts

* [[topological invariance of dimension]]

* [[connected space]]

## References

The original article:

* [[Giuseppe Peano]], *Sur une courbe, qui remplit toute une aire plane*, Mathematische Annalen, **36** 1 (1890) 157–160 [[doi:10.1007/BF01199438](https://doi.org/10.1007/BF01199438)]

See also:

* Wikipedia, _[Peano curve](https://en.wikipedia.org/wiki/Peano_curve)_

* [[Hans Sagan]], *Space-filling curves* (1994) [[doi:10.1007/978-1-4612-0871-6](https://doi.org/10.1007/978-1-4612-0871-6)]

The proof of the Hahn-Mazurkiewicz theorem is given in section 31 (page 219ff) within chapter 8 of Willard's classic text:

* Stephen Willard, *General Topology* (Dover Edition 2004). Originally published by Addison-Wesley, 1970. ([link to vendor](https://www.amazon.com/dp/0486434796/?tag=stackoverfl08-20))

The question of which spaces are continuous images of the real line was asked (and answered with dispatch by Jeff Strom) at MathOverflow:

* [[Todd Trimble]] (https://mathoverflow.net/users/2926/todd-trimble), continuous images of open intervals, URL (version: 2014-05-25): [https://mathoverflow.net/q/168084](https://mathoverflow.net/questions/168084/continuous-images-of-open-intervals)

[[!redirects Peano curves]]
[[!redirects Peano space-filling curve]]
[[!redirects Peano space-filling curves]]
[[!redirects Peano space]]
[[!redirects Peano spaces]]
[[!redirects Hahn-Mazurkiewicz theorem]]


pedigrad

#Contents#
* table of contents
{:toc}

## Idea

In applications of [[category theory]] to [[biology]], a "pedigrad" defines a computational environment to study genetic mechanisms.

## Definition

A _pedigrad_ is a model for a [[limit sketch]] defined on a [category of segments](biology#CategoryOfSegments).

## References

* [[Remy Tuyeras|R Tuyeras]], _Category Theory for Genetics_,  [arXiv:1708.05255](https://arxiv.org/abs/1708.05255)

* [[Remy Tuyeras|R Tuyeras]], _Category theory for genetics I: mutations and sequence alignments_, Theory and Applications of Categories, Vol. 33, 2018, No. 40, pp 1269-1317, [pdf](http://www.tac.mta.ca/tac/volumes/33/40/33-40abs.html5)

* [[Remy Tuyeras|R Tuyeras]], _Category theory for genetics II: genotype, phenotype and haplotype_, [arXiv:1805.07004](https://arxiv.org/abs/1805.07004)

* [[Remy Tuyeras|R Tuyeras]], _A category theoretical argument for causal inference_, [arXiv:2004.09999](https://arxiv.org/abs/2004.09999)


Pedram Hekmati

* [webpage](http://www.maths.adelaide.edu.au/pedram.hekmati/)

## Selected writings

Discussion of [[twisted cohomology|twists]] of [[KR-theory]] by [[HZR-theory]] in degree 3 via [[bundle gerbes]] ([[Jandl gerbes]]) suitable for classifying [[D-brane charge]] on [[orientifolds]]:

* [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], _Real bundle gerbes, orientifolds and twisted KR-homology_, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number  ([arXiv:1608.06466](http://arxiv.org/abs/1608.06466), [doi:10.4310/ATMP.2019.v23.n8.a5](https://dx.doi.org/10.4310/ATMP.2019.v23.n8.a5))

* [[Pedram Hekmati]], [[Michael Murray]], [[Richard Szabo]], [[Raymond Vozzo]], _Sign choices for orientifolds_ ([arXiv:1905.06041](https://arxiv.org/abs/1905.06041))

## Eelated $n$Lab entries

* [[HZR-theory]], [[KR-theory]], [[orientifold]]

category: people


Pedro Boavida de Brito

[Website](https://www.math.tecnico.ulisboa.pt/~pbrito/).

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=199044)

## Selected writings

On [[manifold calculus]] via [[(infinity,1)-sheaves]] on the [[Weiss topology]]:

* {#BriWei} [[Pedro Boavida de Brito]], [[Michael Weiss]], _Manifold calculus and homotopy sheaves_, Homology, Homotopy and Applications, vol. 15(2), 2013, pp.361–383 ([arXiv:1202.1305](http://arxiv.org/abs/1202.1305))

On identifying the [[shape modality]] on the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]] with the [[path ∞-groupoid]]-construction ([[shape via cohesive path ∞-groupoid]]):

* {#BEBP} [[Daniel Berwick-Evans]], [[Pedro Boavida de Brito]], [[Dmitri Pavlov]], _Classifying spaces of infinity-sheaves_ ([arXiv:1912.10544](https://arxiv.org/abs/1912.10544))

## Related concepts

* [[configuration category]]

category: people

[[!redirects Pedro Boavida]]


Pedro Lauridsen Ribeiro

* [webpage](http://professor.ufabc.edu.br/~pedro.ribeiro/)

## related $n$Lab entries

* [[space of sections]]

* [[off-shell Poisson bracket]]

* [[Peierls bracket]]

* [[microcausal polynomial observable]]

category: people

[[!redirects Pedro Ribeiro]]


Pedro Real

* [GoogleScholar page](https://scholar.google.com/citations?user=NmyyYpkAAAAJ)

## Selected writings

On [[Steenrod squares]] and the [[Eilenberg-Zilber/Alexander-Whitney deformation retraction]]:

* {#GonzalezDiazReal99} [[Rocio Gonzalez-Diaz]], [[Pedro Real]], *A Combinatorial Method for Computing Steenrod Squares*, Journal of Pure and Applied Algebra 139 (1999) 89-108 ([arXiv:math/0110308](https://arxiv.org/abs/math/0110308))

category: people


Pedro Resende

__Pedro Resende__ is a mathematician at Instituto Superior Técnico (Lisboa Portugal).
His interests include [[quantale]]s, [[locale]]s, [[groupoid]]s, [[inverse semigroup]]s, [[operator algebras]], sheaf theory, [[topos theory]]...

* [webpage](http://www.math.ist.utl.pt/~pmr)

category: people

[[!redirects P. Resende]]


Pedro Vaz

* [personal page](https://perso.uclouvain.be/pedro.vaz/)

* [GoogleScholar page](https://scholar.google.com/citations?user=K2lhtvcAAAAJ&hl=fr)

## Selected writings

On the [[cobordism category]] of "foams" (from [Khovanov 2004](Khovanov+homology#Khovanov04)) for studying [[link homology]]:

* [[Marco Mackaay]], [[Pedro Vaz]], *The universal $sl_3$-link homology*, Algebr. Geom. Topol. **7** (2007) 1135-1169 [[doi:10.2140/agt.2007.7.1135](https://doi.org/10.2140/agt.2007.7.1135)]

* [[Marco Mackaay]], [[Pedro Vaz]], *The foam and the matrix factorization $sl_3$ link homologies are equivalent*, Algebr. Geom. Topol. **8** (2008) 309-342 [[arXiv:0710.0771](https://arxiv.org/abs/0710.0771), [doi:10.2140/agt.2008.8.309](https://doi.org/10.2140/agt.2008.8.309)]

* [[Marco Mackaay]], [[Pedro Vaz]], *The diagrammatic Soergel category and $sl(N)$-foams, for $N \gt 3$ [[arXiv:0911.2485](https://arxiv.org/abs/0911.2485)]

category: people


Peierls bracket

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#Contents#
* table of contents
{:toc}

## Idea

The _Peierls bracket_ ([Peierls 52](#Peierls52)) is equivalently the canonical [[symplectic manifold|symplectic]] [[Poisson bracket]] on the [[covariant phase space]] of [[field theories]] such as of the [[scalar field]] ([Dütsch-Fredenhagen 01](#DutschFredenhagen01), [Khavkine 14](#Khavkine14), [Collini 16, prop. 31 and below](#Collini16)). Its construction requires the linearized [[field (physics)|field]] [[equations of motion]] satisfied by the [[gauge invariance|gauge invariant]] fields to have unique  [[retarded and advanced Green functions]]. Moreover, it extends "off shell" from the [[phase space]] to the space of all field configurations ([Marolf 94](#Marolf94), [Dütsch-Fredenhagen 03, section 2.1](#DuetschFredenhagen03), [Fredenhagen-Rejzner 12, section 5](#FredenhagenRejzner12)), where however it is only [[Poisson bracket]], no longer [[symplectic manifold|symplectic]]. For more on this aspect see at _[[off-shell Poisson bracket]]_.

The [[Peierls bracket]] of two suitably [[smooth function|smooth]] [[functions]] $A$ and $B$ on [[field (physics)|field]] configuration space is the antisymmetrized influence on $B$ of an [[infinitesimal object|infinitesimal]] perturbation of a [[gauge fixing|gauge-fixed]] [[action]] by a function that restricts to $A$ on the embedding of the space of solutions in the field configuration space. It is the construction of the influence of $A$ on $B$ that requires the existence of unique retarded and advanced [[Green's functions]] of the linearized field equations. One can avoid [[gauge fixing]] the action, as long as $A$ and $B$ are [[gauge invariance|gauge invariant]] [[observables]]. In that case, unique retarded and advanced Green's functions may not exist, but due to the gauge invariance of $A$ and $B$ any representative of the gauge equivalence class of Green's functions with appropriate causal [[support]]. Since gauge invariant observables can be expressed in terms of gauge invariant fields (at least at the linearized level, which is all that matters in the construction) the existence and uniqueness of such equivalence classes of Green's functions is equivalent to the existence and uniqueness of retarded and advanced Green's functions for the linearized field equations satisfied by gauge invariant field combinations.

For example, in [[electrodynamics]], advanced and retarded Green's functions exist on [[globally hyperbolic spacetimes]] for [[Maxwell equations]] in terms of the [[field strength]] $F=\mathbf{d}A$ and correspond to unique gauge equivalence classes of Green's functions for Maxwell equations in terms of the [[vector potential]] $A$.

The [[algebra of functions]] on the space of field configurations becomes a [[Poisson algebra]] in the following way. Pick a set of functions on the space of field configurations that restrict to a non-degeneratce [[coordinate system]] on the embedded [[covariant phase space]]. These functions, together with the [[equations of motion]] and [[gauge fixing]] conditions define a [[Poisson bivector]] by being declared canonical, such that the [[kernel]] of the bivector coincides with the ideal generated by the [[equations of motion]] and the [[gauge fixing]] conditions. Obviously the Poisson structure thus constructed on the algebra of functions on field configurations is not unique and depends on the above choice of coordinates; the same non-uniqueness may be parametrized instead by a choice of a [[connection on a bundle|connection]] on the space of field configurations. The embedded [[covariant phase space]] becomes a [[symplectic leaf]] of the symplectic [[foliation]] of the space of field configurations.

## Definition

### The Peirls bracket

> under construction

+-- {: .num_defn #PeirlsFormula}
###### Definition
(...)

...suitable PDE with advanced/retarded [[Green's function]] $\Delta_S^{A/R}$, then the _causal Gree's function_ is their difference

$$
  \Delta_S \coloneqq \Delta_S^{R} - \Delta_S^A
$$

(...)

=--

([#Khavkine 14, def. 3.9](#Khavkine14))

### The off-shell Peierls bracket

The idea of ([Marolf 93](#Marolf93), section II) is this:

If $\{q(0), p(0)\}$ is the given (symplectic) Poisson bracket
on the space of solutions, identified with the space of
initial data, then requiring that everything Poisson-commutes
with $EL(S)$, the
[[Euler-Lagrange equation|Euler-Lagrange functional]] of the action,
uniquely extends this to a bracket $\{q(t_1), p(t_2)\}$
on all hisories, because commutation with $EL(S)$
involves generation of time translation.
Here $EL(S)$ generates a Poisson ideal and dividing that
out reproduces the original symplectic bracket.

Now observe (Khavkine 1) that $EL(S)$ provides
a foliation of history space by [[symplectic leaves]].
Because under the replacement $EL(S) \mapsto EL(S) - const$
the above still goes through. Observe also (Khavkine 2)
that $EL(S) - const = 0$ is the equations of motion
for the origial action with a [[source]] term $J q$ added.
Hence the off-shell Peierls Poisson structure has
[[symplectic leaves]] parameterized by the [[source]] $J$.

Observe finally (Khavkine 3) that with ([Marolf 93, section III B](#Marolf93)) it follows that the Peierls bracket on the shifted leaves
agrees with the original one. (...)

## Properties

+-- {: .num_prop #PeierlsPoissonBracket}
###### Proposition
**(Peierls bracket is the canonical Poisson bracket in field theory)

Given a [[local Lagrangian field theory]], assume that its [[gauge symmetries]] are globally recognizable ([Khavkine 14, section 3.2.2](#Khavkine14)). Then the Peierls bracket (def. \ref{PeirlsFormula}) is the [[Poisson bracket]] corresponding to the [[symplectic form]] on the [[reduced phase space|reduced]] [[covariant phase space]].

=--

([Khavkine 14, theorem 3.2](#Khavkine14))

## References

The definition of what now is called the Peierls bracket originates in

* {#Peierls52} R. Peierls, _The commutation laws of relativistic field theory_ (1952) ([jstor](http://www.jstor.org/pss/99080))

In this article the Peierls bracket on the [[covariant phase space]] of a [[gauge theory|non-gauge]] system is defined and the equivalence with the Hamiltonian phase space [[symplectic structure]] is (incompletely) demonstrated. Peierls also discusses how the definition extends to gauge theories and to fermionic theories.

An early review is in

* [[Bryce DeWitt]], _The Spacetime Approach to Quantum Field Theory_, in [[Bryce DeWitt]], [[Raymond Stora]] (eds.), _Les Houches Session XL, Relativity, Groups and Topology II_ (North-Holland, 1983), pp. 382–738.

which is also the first to explicitly check the [[Jacobi identity]] for the Peirls bracket.

A streamlined general account is in

* {#Khavkine14} [[Igor Khavkine]], _Covariant phase space, constraints, gauge and the Peierls formula_, Int. J. Mod. Phys. A, 29, 1430009 (2014) ([arXiv.1402.1282](https://arxiv.org/abs/1402.1282))

A traditional physics textbook account amplifying the Peierls bracket is

* {#DeWitt03} [[Bryce DeWitt]], _The global approach to Quantum Field Theory_ (2 volumes), Oxford 2003

The fact that the Peirls bracket for the [[scalar field]] gives the [[Poisson bracket]] in its [[covariant phase space]] is discussed in

* {#DutschFredenhagen01} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative algebraic field theory, and deformation quantization_, in [[Roberto Longo]] (ed.), _Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects_, volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001
([arXiv:hep-th/0101079](https://arxiv.org/abs/hep-th/0101079))

* {#Collini16} [[Giovanni Collini]], prop. 31 and below in _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))

The off-shell generalization to a [[Poisson bracket]] on configuration space (history space) was first given in

* {#Marolf93} [[Don Marolf]],  _Poisson Brackets on the Space of Histories_ Annals of Physics Volume 236, Issue 2, December 1994, Pages 374-391 ([arXiv:hep-th/9308141](http://arxiv.org/abs/hep-th/9308141))

* {#Marolf94} [[Don Marolf]], _The Generalized Peierls Bracket_. Ann. Phys. (N.Y.) 236 (1994) 392–412 ([arXiv:hep-th/9308150](http://arxiv.org/abs/hep-th/9308150))

See also exercise 17.12 in

* [[Marc Henneaux]], [[Claudio Teitelboim]], _[[Quantization of Gauge Systems]]_

A mathematically clean account of the (on- and off-shell) Peierls bracket (for [[scalar fields]]) is in section 2.1 of

* {#DuetschFredenhagen03} [[Michael Dütsch]], [[Klaus Fredenhagen]], _The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory_,  	Commun.Math.Phys. 243 (2003) 275-314 ([arXiv:hep-th/0211242](http://arxiv.org/abs/hep-th/0211242))

and in section 2 of

* Ferdinand Brennecke, [[Michael Dütsch]], _Removal of violations of the Master Ward Identity in perturbative QFT_, Rev.Math.Phys.20:119-172,2008 ([arXiv:0705.3160](http://arxiv-web3.library.cornell.edu/abs/0705.3160))

there with an eye towards the [[renormalization]] program of [[perturbative quantum field theory|perturbative Algebraic Quantum Field Theory]] (pAQFT) on flat and [[AQFT on curved spacetimes|on curved spacetimes]].

* {#DuetschFredenhagen04} [[Michael Dütsch]] and [[Klaus Fredenhagen]], _Causal perturbation theory in terms of retarded products, and a proof of the action Ward identity_, Rev. Math. Phys. 16 (2004) 1291-1348 ([arXiv:hep-th/0403213](http://arxiv.org/abs/hep-th/0403213))

Further references to its use in the renormalization program of pAQFT can be found in:

* {#FredenhagenRejzener12} [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Batalin-Vilkovisky formalism in the functional approach to classical field theory_. Commun.Math.Phys. 314 (2012) 93-127 ([arXiv:1101.5112](http://arxiv.org/abs/1101.5112))

Functional analytic aspects of the definition and existence of the Peierls bracket (including its off-shell extension) are discussed in section 3.2 of

* {#BrunettiFredenhagenRibeiro12} [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Pedro Ribeiro]], _Algebraic Structure of Classical Field Theory I: Kinematics and Linearized Dynamics for Real Scalar Fields_ ([arXiv:1209.2148](http://arxiv.org/abs/1209.2148))

The equivalence between the Peierls bracket and the [[symplectic manifold|symplectic]] [[Poisson bracket]] on the [[covariant phase space]] of classical field theory (by showing the equivalence of both to the canonical Poisson bracket in Hamiltonian formalism) was demonstrated in

* {#BarnichHenneauxSchomblond91} [[Glenn Barnich]], [[Marc Henneaux]], C. Schomblond, _Covariant description of the canonical formalism_. Phys.Rev. D44 (1991) R939-R941 ([doi](http://dx.doi.org/10.1103/physrevd.44.r939))

Further discussion of the manifestly covariant equivalence between the Peierls bracket and the [[symplectic manifold|symplectic]] [[Poisson bracket]] on the [[covariant phase space]] of classical field theory (avoiding traditional proof via the canonical Hamiltonian formalism) can be found in

* [[Michael Forger]], Sandro Romero, _Covariant Poisson Brackets in Geometric Field Theory_, Commun.Math.Phys. 256 (2005) 375-410 ([arXiv:math-ph/0408008](http://arxiv.org/abs/math-ph/0408008))

* [[Igor Khavkine]], _Characteristics, Conal Geometry and Causality in Locally Covariant Field Theory_, section 5, ([arXiv:1211.1914](http://arxiv.org/abs/1211.1914))

[[!redirects Peierls brackets]]

[[!redirects Peierls-Poisson bracket]]
[[!redirects Peierls-Poisson brackets]]

[[!redirects Poisson-Peierls bracket]]
[[!redirects Poisson-Peierls brackets]]


Peirce > history

[[< history]]


Peirce's law

Peirce's law is a [[proposition]] that involves only [[conditionals]] and is valid in [[classical logic]] but not [[intuitionistic logic]]:

$$ ((p \Rightarrow q) \Rightarrow p) \Rightarrow p $$

It is named after [[Charles Sanders Peirce]].

[[!redirects Peirce's axiom]]
[[!redirects Peirce's Axiom]]
[[!redirects Peirce's Law]]
[[!redirects Peirce law]]
[[!redirects Peirce's law]]


Peirce's logic of information

The material previously on this page is now located at the [Online Encyclopedia of Integer Sequences Wiki](https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension).

[[!redirects ICE]]
[[!redirects PLOI]]
[[!redirects I = C × E]]
[[!redirects information = comprehension × extension]]
[[!redirects Information = Comprehension × Extension]]
[[!redirects Peirce's logic of information]]


Pekka J. Lahti

* [institute page](https://www.utu.fi/en/people/pekka-lahti)

* [InSpire page](https://inspirehep.net/authors/2300690)

## Selected writings

On [[quantum physics]] with emphasis on [[quantum probability]] via [[POVMs]]:

* [[Paul Busch]], Marian Grabowski, [[Pekka J. Lahti]], *Operational Quantum Physics*, Lecture Notes in Physics Monographs **31**, Springer (1995) [[doi:10.1007/978-3-540-49239-9](https://doi.org/10.1007/978-3-540-49239-9)]

* [[Paul Busch]], [[Pekka J. Lahti]], Juha-Pekka Pellonpää, Kari Ylinen, *Quantum Measurement*, Springer (2016) [[doi:10.1007/978-3-319-43389-9](https://doi.org/10.1007/978-3-319-43389-9)]

category: people

[[!redirects Pekka Lahti]]


Pelle Steffens

* [institute page](https://www.math.cit.tum.de/en/algebra/personen/dr-pelle-steffens/)

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=288580)

## Selected writings

On [[derived smooth manifolds]]:

* {#CarchediSteffens19} [[David Carchedi]], [[Pelle Steffens]], _On the universal property of derived manifolds_ [[arXiv:1905.06195](https://arxiv.org/abs/1905.06195)]

* [[Pelle Steffens]], *Derived $C^\infty$-Geometry I: Foundations* [[arXiv:2304.08671](https://arxiv.org/abs/2304.08671)]

category: people


pencil

## Rough idea

A pencil is a 1-parametric family of divisors in algebraic geometry.

## Definition

A _pencil_ is a linear system of [[divisor (algebraic geometry)|divisor]]s of dimension $1$, that is parametrized by a projective line.

## Examples

#### Pencil of lines through a given point

The basic example is the _pencil of all lines_ passing through a single point in a projective plane, or in an affine plane.

The pencil of lines which pass through a point at infinity in projective plane consists from the point of view of an affine complement of the line at infinity to an equivalence class of parallel lines.

Conversely the projective completion of any affine space is obtained by adding points at infinity. Each point can be identified with the set of all lines through it (this is related to the projective duality). A point at infinity attached to an affine space is defined as the equivalence class of parallel lines (see [[direction of a line]]) which may be viewed as those lines in the affine space which are pointing toward that point at infinity.

#### Pencils of conics

...


Penelope Maddy

* [institute page](https://faculty.sites.uci.edu/pjmaddy/)

* [Wikipedia](https://en.wikipedia.org/wiki/Penelope_Maddy)

## Selected publications

* [[Penelope Maddy]], *What Do We Want a Foundation to Do? Comparing Set-Theoretic, Category-Theoretic, and Univalent Approaches*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019)  [[doi:10.1007/978-3-030-15655-8_13](https://doi.org/10.1007/978-3-030-15655-8_13), [pdf](https://bpb-us-e2.wpmucdn.com/faculty.sites.uci.edu/dist/3/399/files/2023/09/What-do-we-want-a-foundation-to-do.pdf)]

## Related concepts

* [[philosophy of mathematics]]

* [[foundations of mathematics]]

category: people


Peng Fu

* [webpage](https://fermat.github.io/)

## Selected writings

On [[quantum programming]] via [[dependent linear type theory]]:

* {#FKS20} [[Peng Fu]], [[Kohei Kishida]], [[Peter Selinger]], _Linear Dependent Type Theory for Quantum Programming Languages_, LICS '20: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer ScienceJuly 2020 Pages 440–453 ([arXiv:2004.13472](https://arxiv.org/abs/2004.13472), [doi:10.1145/3373718.3394765](https://dl.acm.org/doi/10.1145/3373718.3394765), [pdf](https://depend.cs.uni-saarland.de/lics-icalp/papers/B5.F), [video](https://www.youtube.com/watch?v=GUT8j4V6Nzg))

On [[quantum programming]] via [[dependent linear type theory]] with [[Quipper]]:

* [[Peng Fu]], [[Kohei Kishida]], [[Neil Ross]], [[Peter Selinger]],  _A Tutorial Introduction to Quantum Circuit Programming in Dependently Typed Proto-Quipper_, in I. Lanese, M. Rawski  (eds.) _Reversible Computation_ RC 2020. Lecture Notes in Computer Science, vol 12227 ([arXiv:2005.08396](https://arxiv.org/abs/2005.08396), [doi:10.1007/978-3-030-52482-1_9](https://doi.org/10.1007/978-3-030-52482-1_9))

On [[dependent linear type theory]] and [[categorical semantics]] for versions of "Proto-[[Quipper]]":

* [[Peng Fu]], [[Kohei Kishida]], [[Neil J. Ross]], [[Peter Selinger]], *A biset-enriched categorical model for Proto-Quipper with dynamic lifting* [[arXiv:2204.13039](https://arxiv.org/abs/2204.13039)]

* [[Peng Fu]], [[Kohei Kishida]], [[Neil J. Ross]], [[Peter Selinger]], *Proto-Quipper with dynamic lifting* [[arXiv:2204.13041](https://arxiv.org/abs/2204.13041)]

* [[Frank (Peng) Fu]], *Proto-Quipper with Dynamic Lifting*, [talk at](CQTS#PengFuOct2023) [[CQTS]] (23 Oct 2023) [video:[YT](https://youtu.be/bBL7rlqbDWM)]

category: people

[[!redirects Frank Fu]]

[[!redirects Frank (Peng) Fu]]


Peng Ye

* [personal page](https://sites.google.com/site/yphysics/)

* [GoogleScholar page](https://scholar.google.ca/citations?user=E-4DSAEAAAAJ)

## Selected writings

On [[symmetry protected topological phases]]:

* {#YeWen13} [[Peng Ye]], [[Xiao-Gang Wen]], *Projective construction of two-dimensional symmetry-protected topological phases with $\mathrm{U}(1)$, $SO(3)$, or $SU(2)$ symmetries*, Phys. Rev. B **87** 195128 (2013) $[$[doi:10.1103/PhysRevB.87.195128](https://doi.org/10.1103/PhysRevB.87.195128), [arXiv:1212.2121](https://arxiv.org/abs/1212.2121)$]$

On [[loop braid group]]-statics in (models for) [[topological phases of matter]]:

* AtMa P. O. Chan, [[Peng Ye]], [[Shinsei Ryu]], *Braiding with Borromean Rings in (3+1)-Dimensional Spacetime*, Phys. Rev. Lett. 121, 061601 (2018) [[arXiv:1703.01926](https://arxiv.org/abs/1703.01926)]

* Zhi-Feng Zhang, Qing-Rui Wang, [[Peng Ye]], *Wilson Operators, Non-Abelian Fusion Rules, and Loop Shrinking Rules of Borromean Rings Topological Order* [[arXiv:2208.09228](https://arxiv.org/abs/2208.09228)]

category: people


Penrose diagram

> This entry is about the [[Roger Penrose]]'s diagrams reflecting [[causal structure]] of [[spacetimes]] in [[general relativity]]. For the [[Penrose notation]] for [[tensors]] in [[linear algebra]] see at _[[string diagram]]_.

***

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{:toc}

## Idea

In the theory of [[gravity]] ([[general relativity]]) the _Penrose diagram_ of a given [[spacetime]] is a 2-dimensional depiction ([[time]] running vertically and a single spatial distance coordinate running horizontally) of its [[causal structure]], reflecting the projected image of the spacetime under a [[conformal transformation]] which makes [[events]] at [[timelike]]/[[spacelike]]/[[lightlike]] infinity of the spacetime appear at respective [[boundary]] points of its Penrose diagram.

## References

Due to [[Roger Penrose]].

See also:

* Wikipedia, _[Penrose diagram](https://en.wikipedia.org/wiki/Penrose_diagram)_

[[!redirects Penrose diagrams]]


Penrose singularity theorem

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## Idea

The _Penrose-Hawking singularity theorems_ characterize [[spacetimes]] in the theory of [[Einstein]]-[[gravity]] ([[general relativity]]) which have "singularities", points where the [[Riemann curvature]] is undefined (or would be undefined if these points were included in the spacetime [[manifold]]) such as appears notably in [[black hole spacetimes]].

[Hellman](#Hellman) suggest this theorem as an example of non[[computable physics]]. See [Frank](#Frank) for a response.

A related problem is that of the [maximal Cauchy development for the Einstein equations](#Sbierski). In this case, at least [[Zorn's lemma]] can be avoided.

## Related concepts

* [[conformal compactification]]

* [[cosmic censorship hypothesis]]

## References

The original article:

* [[Stephen Hawking]], [[Roger Penrose]], _The Nature of Space and Time_ Princeton: Princeton University Press. ISBN 0-691-03791-4. (1996)

Review:

* [[Klaas Landsman]], _Singularities, black holes, and cosmic censorship: A tribute to Roger Penrose_ ([arXiv:2101.02687](https://arxiv.org/abs/2101.02687))

* [[José M. M. Senovilla]], *A critical appraisal of the singularity theorems* ([arXiv:2108.07296](https://arxiv.org/abs/2108.07296))

See also:

* Wikipedia, _[Penrose-Hawking singularity theorems](https://en.wikipedia.org/wiki/Penrose%E2%80%93Hawking_singularity_theorems)_

For further developments see

* [[Alan Coley]], _Mathematical General Relativity_ ([arXiv:1807.08628](https://arxiv.org/abs/1807.08628))

In view of [[constructive mathematics]]:

* {#Hellman} Geoffrey Hellman, _Mathematical constructivism in spacetime_, British Journal for the Philosophy of Science 49 (3):425-450 (1998) [PDF](http://www.jstor.org/stable/688083)

* {#Frank} Matthew Frank, _Axioms and aesthetics in constructive mathematics and differential geometry_. PhD-thesis, Chicago, 2004.

* {#Sbierski} Jan Sbierski, _On the Existence of a Maximal Cauchy Development for the Einstein Equations - a Dezornification_ [PDF](http://arxiv.org/abs/1309.7591)

Cautioning that the singularity theorem makes statements about non-extendible curves but *not* explicitly about "singularities" in the sense of points of diverging curvature:

* [[Roy Kerr]], *Do Black Holes have Singularities?* [[arXiv:2312.00841](https://arxiv.org/abs/2312.00841)]

Informed commentary is made in reply to [Physics.SE:q/790724](https://physics.stackexchange.com/q/790724/5603), especially  [Physics.SE:a/796154](https://physics.stackexchange.com/a/796154/5603):

> Penrose's theorem guarantees, under certain hypotheses, that spacetime is null geodesically incomplete. [...] The primary point of Kerr's paper is that this theorem has nothing to do with the central ring singularity of his namesake spacetime. The theorem does not tell us that the ring singularity is there: the singularity could, in principle, be excised from the spacetime and replaced with a self-supporting stationary matter distribution without contradicting the theorem in any way.

[[!redirects Penrose singularity theorems]]

[[!redirects Penrose-Hawking singularity theorem]]

[[!redirects Penrose-Hawking singularity theorems]]

[[!redirects Penrose-Hawking theorem]]


Penrose tiling

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Properties

* [[noncommutative topology of quasiperiodicity]]

## Related entries

* [[tesselation]]

* [[quasi-crystal]]

## References

* Wikipedia, *[Penrose tiling](https://en.wikipedia.org/wiki/Penrose_tiling)*

In relation to [[quantum error correcting codes]]:

* Zhi Li, [[Latham Boyle]], *The Penrose Tiling is a Quantum Error-Correcting Code* [[arXiv:2311.13040](https://arxiv.org/abs/2311.13040)]

category: people

[[!redirects Penrose tilings]]


pentagon

#Contents#
* table of contents
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## Idea

The *pentagon* is any [[polygon]] with 5 sides.

## Related concepts

* [[pentagon identity]]

## References

See also:

* Wikipedia, *[Pentagon](https://en.wikipedia.org/wiki/Pentagon)*
* Dragutin Svrtan, Darko Veljan, Vladimir Volenec, _Geometry of pentagons: from Gauss to Robbins_, [arXiv:math.MG/0403503](https://arxiv.org/abs/math/0403503)

[[!redirects pentagons]]


pentagon decagon hexagon identity

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#### Geometry
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#Contents#
* table of contents
{:toc}

#Idea#

A certain identity in [[Euclidean geometry]].

Suppose we inscribe a regular pentagon, a regular decagon, and a regular hexagon in circles of the same radius.  If we denote the respective edge lengths of these polygons by $P$, $D$ and $H$, then these lengths will satisfy the identity:

$$P^2=D^2+H^2$$

This means that the edges of a pentagon, decagon and hexagon of identical radii can fit together to form a right triangle.

Euclid states this beautiful but mysterious identity as [Proposition 10](http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII10.html) of Book XIII of the _Elements_.  This is the last book of the _Elements_, the one which deals with properties of the Platonic solids.  He uses Proposition 10 as part of his construction of the regular icosahedron in [Proposition 13](http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII13.html).

This has led various historians to suggest that the pentagon-decagon-hexagon identity was first discovered in the course of research on the icosahedron.  However, it seems that until now there has been no _proof_ of the pentagon-decagon-hexagon identity based on properties of the icosahedron.  We present such a proof here.

We also present two other proofs: a modernized version of Euclid's original proof, and a somewhat simpler proof which (like Euclid's) uses only 2-dimensional geometry.

#Euclid's proof#

The essence of [Euclid's proof](http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII10.html) can be understood if we take for granted the numerical values of the various angles in his construction, which are easily established.  Euclid himself would not have used numerical values like this, and they are not a part of the proof; they're merely a short-hand that allows the reader to see at a glance the relationships between angles that the original proof established by routine methods.

[[!include pentagon decagon hexagon identity > Euclid]]

The line segment $A B$ is one edge of a pentagon inscribed in a circle centred at $F$, and $A K$ and $K B$ are edges of a decagon inscribed in the same circle.  Note that while $A B$ is bisected by $H$ and $A K$ is bisected by $L$, the point $N$ is _not_ a point of bisection of $A H$.

Triangle $A B F$ is similar to triangle $B F N$.  So $\frac{A B}{B F}=\frac{B F}{F N}=\frac{B F}{B N}$, with the last equality true because the triangles are isosceles with $F N = B N$.  Thus ${B F}^2={A B}\cdot{B N}$.

Triangle $B A K$ is similar to triangle $K A N$.  So $\frac{B A}{A K}=\frac{K A}{A N}$.  Thus ${A K}^2={A B}\cdot{A N}$.

Adding our two results, we have:
${B F}^2+{A K}^2={A B}\cdot(A N + B N)={A B}^2$.

$B F$ is our radius (or equivalently, the length of one side of a hexagon inscribed in the same circle), $A K$ is a decagon side, and $A B$ is a pentagon side.  So we have established the pentagon-decagon-hexagon identity.

#Another 2d proof#

Euclid's proof is quick but somewhat mysterious.  Here is another, perhaps simpler, proof that uses only 2-dimensional constructions. It relies on a lemma concerning the ratio of sides in a "golden triangle".

##Golden triangle lemma##

[[!include pentagon decagon hexagon identity > golden triangle]]

The lemma states that in any isosceles triangle with angles of 72°, 36° and 72°, the ratio of the long sides of the triangle to its base is $\Phi$, the golden ratio, which is defined as the unique positive solution of the equation:

$$1+\frac{1}{\Phi}=\Phi$$

Note that some authors call "the golden ratio" the reciprocal of the number we're calling $\Phi$.  For the purpose of this article, $\Phi=\frac{\sqrt{5}+1}{2}\approx 1.61803$.

In the diagram above, triangle $B D E$ is similar to triangle $B E C$.  So:

$$\frac{B D}{B E}=\frac{B E}{B C}$$
$$\frac{r+t}{r}=\frac{r}{t}$$
$$1+\frac{t}{r}=\frac{r}{t}$$

By the definition of the golden ratio, $\frac{r}{t}=\Phi$.

This lemma was given by Euclid as [Proposition 9](http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII9.html) of Book XIII.

##Proof using the golden triangle lemma##

[[!include pentagon decagon hexagon identity > golden ratio]]

By the golden triangle lemma for isosceles triangles with angles of 36°, triangle $N O P$ gives us $r=t\Phi$.  From the same lemma, applied to triangle $P N Q$, we have $2b=t/\Phi$.  So the product of $r$ and $2b$ is $t^2$.

Because triangle $N Q O$ and $N Q P$ are both isosceles, $O Q = N Q = N P = t$, the decagon edge.  And we can read off the equation $r=t+2b$ just by summing the distances along $O P$.

From the right triangle $N M P$ we have $s^2=t^2-b^2$, hence $(2s)^2=4t^2-4b^2$.

The whole large multi-coloured rectangle has an area of $(2s)^2$, because the green square, the light blue rectangle, and each of the two red rectangles _all_ have areas of $t^2$, but then we need to subtract the light red
square where the two red rectangles overlap, and that has area $4b^2$.

Now if we subtract a single $t^2$, the light blue rectangle, we're left with a square of side length $r$.  So $(2s)^2-t^2=r^2$, which is the pentagon-decagon-hexagon identity.

###Alternative construction displaying right triangles###

While the construction above is straightforward, it's also possible to present an equivalent result that explicitly gives us right triangles with the pentagon edge as their hypotenuse and the hexagon and decagon edges as their other sides.

[[!include pentagon decagon hexagon identity > pythagorean identity]]

The diagram above is similar to one used in a common proof of Pythagoras's Theorem, but here we use it to establish the value of the square of the hypotenuse of a right triangle with sides $r$ and $t$, i.e. the hexagon edge and the decagon edge.

The large tilted square has a hypotenuse as its side.  Its area is $4b^2$, from the central pink square, plus $4t^2$, from the four multi-coloured $t\times t$ squares arranged around the central square, minus four times the area of the dark green triangle that needs to be excluded (the contributions from the yellow triangles inside and outside the region cancel out).  But each dark green triangle has both base and height of $2b$, and hence an area of $2b^2$, so the total area is $4t^2+4b^2-4\times 2b^2=4t^2-4b^2=(2s)^2$, the square of a pentagon edge.

At first glance it might seem as if we've only used $r=t+2b$ to derive the result here, whereas in the previous construction we needed the fact that the product of $r$ and $2b$ was $t^2$.  But we _have_ used it:  the yellow triangles that share an edge with the dark green triangles, and those that do not, are only congruent because of it.  The latter have one of their sides by construction being $r-t=2b$, whereas the former's matching side arises as a proportion of the larger $r\times t$ triangle as $\frac{t}{r}\cdot t=2b$.

##An identity in the golden ratio##

The first way of presenting the preceding proof implies the identity:

$$\Phi^2+\frac{1}{\Phi^2}=3$$

which can also be verified algebraically from the defining equation for $\Phi$.  We can see this identity a bit more clearly if we remove the leftmost rectangle from the construction, slide the rightmost one down to eliminate the overlap, and take out the overall factor of $t$ from all the sides.  We then have:

[[!include pentagon decagon hexagon identity > golden ratio identity]]

We can also construct a diagram that explicitly shows that the square of the diagonal of a rectangle of dimensions $\Phi$ by $\frac{1}{\Phi}$ is $3$.

[[!include pentagon decagon hexagon identity > golden ratio identity pythagorean]]

Here each red triangle has area $\frac{1}{2}$, so along with the light green unit square the total area is $3$.

#Proof using the icosahedron#
{. #Icosahedron}

Suppose we take **three congruent rectangles**, and assemble them symmetrically so that their _long_ edges are all mutually orthogonal, and their _short_ edges are all mutually orthogonal.  Like this:

[[!include pentagon decagon hexagon identity > skinny duads]]

We then join up the corners to make an icosahedron, as in the diagram above.  Eight faces of this icosahedron (the ones centred on the diagonals of the eight "octants" into which the space here is divided) will _always_ be congruent and equilateral; that's guaranteed by the construction, whatever the precise ratio of the side lengths of the rectangles.  The other twelve faces will be isosceles triangles, all congruent to each other.

In terms of edges, there are two congruent sets:  the six edges that correspond to short sides of our rectangles will all be congruent, and the twenty-four edges that are sides of the eight equilateral faces will all be congruent.

Now, it's clear that by making the rectangles long and skinny, we can make the isosceles triangles as long and skinny as we like.  But what about making them short and squat?  How far can we go in that direction?

[[!include pentagon decagon hexagon identity > square duads]]

If we make the three rectangles into squares, as above, the isosceles triangles have two 45° angles.  So between this extreme, and the long-and-skinny extreme, it's clear that we can choose _some_ intermediate degree of skinniness for the rectangles that will turn the 12 isosceles triangles into equilateral triangles.

So, a _regular_ icosahedron must exist, and it will have six edges that form three mutually orthogonal pairs.

[[!include pentagon decagon hexagon identity > duads]]

(The argument we've just given implicity uses the [[intermediate value theorem]] and would not have been used by Euclid, although Archimedes might have attempted it.  An alternative that avoids using the intermediate value theorem is given at the end of the proof.)

We will set the short side of the rectangles to $1$, and for now we will call the corresponding value for the long side that produces a regular icosahedron $x$.

In the figure below we project everything into the plane of one of the rectangles.  Where two vertices are projected to the same point in the diagram, a circle is drawn around the point.

[[!include pentagon decagon hexagon identity > icosahedron]]

The five vertices of the icosahedron that project onto the line segment $A P R$ will form a regular pentagon, the edges of which are also edges of the icosahedron.  $A P R$ is a straight line because the pentagon is a planar figure, as it must be if it forms the base of a pentagonal pyramid with five congruent equilateral triangles as its other faces.  The point $C$ is produced by dropping a perpendicular line from the apex, $B$, of this pentagonal pyramid, to its base.

By construction, $Q R$ is parallel to $A B$, and both are orthogonal to $O P$.  Combined with the fact that $A P R$ is a straight line, that's enough to show that the triangle $A B C$ is congruent to the triangle $S P T$.

The five vertices that project onto the line segment $Z S Q$ will form another pentagon.  The two pentagons will be congruent, and the radius of the circles in which they're inscribed will be $A C$.  And since the triangle $A B C$ is congruent to the triangle $S P T$, the _distance_ between the pentagons, $S T$, equals their radius, $A C$.

These two pentagons are shown in the diagram below.

[[!include pentagon decagon hexagon identity > icosahedron triangles]]

If we draw a line perpendicular to the planes of the two pentagons, from a vertex in one plane, $A$, to the point directly below it on the other plane, $V$, the length of that perpendicular, $A V$, will be equal to the common radius of the pentagons, $A C$.  Now, $A V$ is one side of a right triangle whose hypotenuse is an icosahedral edge, $A Q$ (congruent to the edges of the pentagons), and whose third side, $Q V$, is one edge of a decagon inscribed in the same circle as the lower pentagon.  The decagon edge arises because the two pentagons are rotated relative to each other by an angle of 36°, the angle between vertices in a decagon.

So, the triangles $A B C$ and $A Q V$ are congruent (both being right triangles with a pentagon edge as their hypotenuse and the radius as one of their sides), and both exhibit the pentagon-decagon-hexagon identity.

According to the historian Ian Mueller, Eva Sachs, in her book _Die Fünf Platonischen Körper_, suggested an accurately drawn figure could let someone guess that the distance between the two pentagons equals the radius of either one.  Mueller also writes that Dijksterhuis (in 1929) and Neuenschwander (in 1975) claimed it's "intuitively evident" that the two right triangles above are congruent.   These clues led Greg Egan to the present proof.

If we want, we can now appeal to Euclid's golden triangle lemma -- which gives us the ratio between the radius and the decagon edge length as $\Phi$ -- and work backwards to show that this is also the ratio of length to width for our three orthogonal rectangles; that is, $x=\Phi$ is the value needed to produce a regular icosahedron.  This follows simply from the fact that the right triangle $O R P$ is similar to the right triangle $C A B$.

##Avoiding the intermediate value theorem##

Finally, suppose we wish to avoid using the intermediate value theorem.  To do this, we need to specify the length-to-width ratio, $x$, of the rectangles in our construction, and show that our choice yields a regular icosahedron.

One approach we could take is to set $x=\Phi$, and directly calculate the length of an edge of our icosahedron that doesn't lie on one of the rectangles.  If this length is $1$ -- matching the edges that _do_ lie on the rectangles -- the icosahedron will be regular.

[[!include pentagon decagon hexagon identity > pythagorean icosahedron]]

As the diagram above shows, this isn't too hard; with the help of a re-scaled version of the diagram we used to demonstrate the identity $\Phi^2+\frac{1}{\Phi^2}=3$, we can see that ${A Q}^2=\frac{3}{4}$, and, since the vertices that project to $Q$ are a distance of $\frac{1}{2}$ out of the plane, the squared length of the edge that projects to $A Q$ will be $1$.

But if this is a little too close to resorting to algebra, we can take a more geometric approach.

If we project the generic version of our icosahedron onto the plane of one of its rectangles, we get this:

[[!include pentagon decagon hexagon identity > irregular icosahedron]]

In order to make $A P R$ into a straight line -- which amounts to making the pentagon of vertices that project onto $A P R$ into a planar figure -- we need the slope of $A R$ to equal the slope of $P R$, which is equivalent to:

$$\frac{x+1}{x}=x$$

This is just the defining equation for the golden ratio, $\Phi$.  So we need to set $x=\Phi$ to have any hope of our icosahedron being regular, since in a regular icosahedron the pentagon will be the base of a pentagonal pyramid with equilateral sides, which is clearly a planar figure.

Next, we note that the golden triangle lemma can be used to establish that in the figure below, the larger of the two golden triangles here associated with a regular pentagon is $\Phi$ times larger than the smaller one (because the long edge of the smaller triangle is congruent to the small edge of the larger triangle).  This means the _height_ of a regular pentagon, measured from one edge, is $\Phi$ times the height of an adjacent vertex.

[[!include pentagon decagon hexagon identity > pentagon heights]]

The same lemma also shows that the _width_ of a regular pentagon, measured parallel to one edge, is $\Phi$ times its edge length.

[[!include pentagon decagon hexagon identity > pentagon width]]

Now, by choosing $x=\Phi$, as well as making the pentagon that projects onto $A P R$ into a planar figure, we've actually made _every_ set of five vertices that are neighbours to any given vertex in the icosahedron lie in a plane.

[[!include pentagon decagon hexagon identity > icosahedron pentagon]]

For example, the two such pentagons that project onto $B P S Z R$ must also be planar.  But since they're planar, their projection here will preserve the ratio between the perpendicular distance from $P S$ to the vertex that projects to $R$ and the perpendicular distance from $P S$ to the vertices that project to $Z$ and $B$.  The projections of these distances are just $\frac{x}{2}$ and $\frac{1}{2}$ respectively, so our choice of $x=\Phi$ makes their ratio equal to $\Phi$.

Furthermore, the lengths of the line segments projecting to $B Z$ and $P S$ will be the same in the actual pentagons as they are in the projection, and in the projection their ratio is $\frac{B Z}{P S}=x=\Phi$.

But we've established that both these ratios are $\Phi$ for a _regular_ pentagon, and the pentagons projecting to $B P S Z R$ are already so symmetrical (with at least four congruent edges, and mirror symmetry in the perpendicular bisector of the fifth edge) that if they share both ratios, they too must be regular.

[[!include pentagon decagon hexagon identity > regular pentagon]]

That's probably intuitively obvious, but we can make it a bit more explicit.  In the diagram above, the blue pentagon shares one edge with the regular, black pentagon, and has also been constrained to have the same width and height ratios.  The height ratio fixes $\frac{J K}{G H}=\Phi$ and the width ratio fixes $\frac{F L}{E F}=\Phi$, which means the two yellow triangles are similar, which in turn fixes $\frac{F K}{F H}=\Phi=\frac{F K'}{F H'}$.

If we further require that the triangle $F K H$ is isosceles (which follows from the way we've constructed the generic icosahedron), then it will be similar to $F K' H'$, and $F L K$ will be similar to $F L K'$ (since the angle of either of the two yellow triangles at $F$ is just half the supplement of the angle $K F H$).  But since $F L K$ and $F L K'$ share an edge, they will actually be _congruent_, and the blue and black pentagons will coincide exactly.

In other words, there is only one pentagon meeting all the conditions we've imposed:  the regular pentagon with edge $F L$.  This means that, given our choice of $x=\Phi$, the two kinds of congruent edges in our generic icosahedron end up being the same size, and the icosahedron itself is regular.

#A dual proof using the icosahedron#
{. #IcosahedronDual}

The pentagon-decagon-hexagon identity is usually demonstrated by exhibiting a right triangle whose sides are the edges of a regular pentagon, decagon and hexagon _with the same radii_.  However, a proof of the identity devised by Ian Agol constructs a pentagon, decagon and hexagon associated with the icosahedron that all share a common edge, and whose _radii_ (now of different lengths) correspond to the _altitudes_ of a right triangle.

If the length of the common edge of the three polygons is taken to be 1, then their radii are the _reciprocals_ of the edge lengths $P$, $D$ and $H$ of the same polygons if they had radii of 1.  But the altitudes of a right triangle obey a "reciprocal" or "dual" version of the usual Pythagorean theorem:  if the altitudes of a right triangle measured from its two perpendicular sides are $a$ and $b$, and that measured from the hypotenuse is $c$, then it is easily shown that:

$$\frac{1}{c^2} = \frac{1}{a^2} + \frac{1}{b^2}$$

It follows that the existence of a right triangle with altitudes $a=\frac{1}{D}$, $b=\frac{1}{H}$ and $c=\frac{1}{P}$ implies the pentagon-decagon-hexagon identity:

$$P^2=D^2+H^2$$

[[!include pentagon decagon hexagon identity > agol icosahedron]]

In the diagram above, we have chosen a vertex $V$ of a regular icosahedron with center $C$.  The points $M_1$ and $M_2$ are the midpoints of two edges of the icosahedron incident on $V$, and we construct three regular polygons that all have $M_1 M_2$ as a common edge.

A regular hexagon (blue) is constructed with center $V$, with $M_1 M_2$ as one of its edges, lying in the plane of one of the faces of the icosahedron.

A regular pentagon (yellow) is constructed, with its five vertices being the midpoints of the five edges of the icosahedron that are incident on $V$.  The center of this pentagon will lie on the line segment $C V$, and is named $W$ in the diagram.  The plane in which this pentagon lies is orthogonal to $C V$, so the line segment $W M_1$ meets $C V$ in a right angle.

A regular decagon (green) is constructed with center $C$, with $M_1 M_2$ as one of its edges.

Finally, a right triangle (red) is constructed, with vertices $C$, $M_1$ and $V$.  The angle at $M_1$ is a right angle, because $C M_1$ is a perpendicular bisector of the polyhedron edge of which $M_1$ is the midpoint.  The altitude $C M_1$ of this triangle is the radius of the decagon.  The altitude $V M_1$ is the radius of the hexagon.  And the altitude $W M_1$ is the radius of the pentagon.  From the dual Pythagorean theorem, this establishes the pentagon-decagon-hexagon identity.

Ian Agol's construction can also be applied to any of the other Platonic solids, and also to the four Kepler-Poinsot polyhedra.  This yields the following results:

From the tetrahedron and from the cube, $T^2=S^2+H^2$ for a triangle, square and hexagon.

From the octahedron, $S^2=H^2+H^2$, for a square and two hexagons.

From the dodecahedron, $T^2=D^2+D_3^2$, for a triangle, decagon, and a 10-pointed star polygon, $D_3$, whose ten edges subtend a total of 1080°.  This star polygon arises as the analogous polygon to the blue hexagon in the icosahedral case.  Its center is a vertex $V$ of the dodecahedron, and it has as one edge the line segment joining the midpoints of two adjacent edges of a pentagonal face (both edges being incident on the vertex $V$).  Since the interior angle of a regular pentagon at each corner is 108°, the corresponding polygon must be a star polygon with its edges subtending that angle, and the first multiple of 108° that is also a multiple of 360° is 1080°, from ten edges.

[[!include pentagon decagon hexagon identity > agol dodecahedron]]

Applying the construction to the Kepler-Poinsot polyhedra, the great dodecahedron and the great icosahedron both yield $P_2^2=H^2+D_3^2$, where $P_2$ is a pentagram.  The small stellated dodecahedron yields the original identity, $P^2=D^2+H^2$, with the locations of the hexagon and decagon swapped compared to the icosahedral construction.  And the great stellated dodecahedron yields $T^2=D^2+D_3^2$, the same identity as the ordinary dodecahedron, with the decagon and the 10-pointed stars changing places.

In general, one of the polygons (the "green" one) is invariant under taking the dual of the polyhedron.  If the faces of the polyhedron are $n$-gons and the vertex figures are $v$-gons, the "yellow" polygon  (whose vertices are all the midpoints of all the edges incident on some vertex of the polyhedron) will be a $v$-gon, with each edge subtending an angle of $\frac{2\pi}{v}$, while the "blue" polygon centred on the vertex will be a $\frac{2n}{n-2}$-gon, where each edge subtends an angle of $\frac{(n-2)\pi}{n} = \pi-\frac{2\pi}{n}$ (the interior angle between the edges of an $n$-gon).

The difference of the squares of the sines of half these angles is:

$$\sin\left(\frac{\pi}{v}\right)^2 - \sin\left(\frac{(n-2)\pi}{2n}\right)^2 = -\cos\left(\pi \left(\frac{1}{n}+\frac{1}{v}\right)\right) \cos\left(\pi \left(\frac{1}{n}-\frac{1}{v}\right)\right)$$

where the RHS is now clearly invariant under an exchange of $n$ and $v$.  This quantity is the squared sine of half the angle subtended by each edge of the "green" polygon.

#References#

* John Baez, [This Week's Finds in Mathematical Physics, Week 283](http://math.ucr.edu/home/baez/week283.html)

* [n-Café discussion of TWF 283](http://golem.ph.utexas.edu/category/2009/11/this_weeks_finds_in_mathematic_44.html)

* John Baez, [Some Thoughts on the Number 6](http://math.ucr.edu/home/baez/six.html)

* Ian Mueller, _Philosophy of Mathematics and Deductive Structure in Euclid's Elements_, MIT Press, Cambridge Massachusetts, 1981, pp. 257--258 and references therein.

* Eva Sachs, _Die Fünf Platonischen Körper, zur Geschichte der Mathematik und der Elementenlehre Platons und der Pythagoreer_, Berlin, Weidmann, 1917, pp. 102--104.
See page 102--103 [here](http://math.ucr.edu/home/baez/icosahedron_sachs_1.jpg) and page 104 [here](http://math.ucr.edu/home/baez/icosahedron_sachs_2.jpg).

[[!redirects pentagon-decagon-hexagon identity]]
[[!redirects pentagon–decagon–hexagon identity]]
[[!redirects pentagon--decagon--hexagon identity]]


pentagon decagon hexagon identity > agol dodecahedron

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pentagon decagon hexagon identity > agol icosahedron

[[!redirects pentagon decagon hexagon identity/agol icosahedron]]
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pentagon decagon hexagon identity > duads

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