Huzita–Hatori axioms

Details

Axiom 1

Given two points p₁ and p₂, there is a unique fold that passes through both of them.

In parametric form, the equation for the line that passes through the two points is :

${\displaystyle F(s)=p_1+s(p_2-p_1).}$

Axiom 2

Given two points p₁ and p₂, there is a unique fold that places p₁ onto p₂.

This is equivalent to finding the perpendicular bisector of the line segment p₁p₂. This can be done in four steps:

• Use Axiom 1 to find the line through p₁ and p₂, given by ${\displaystyle P(s)=p_1+s(p_2-p_1)}$
• Find the midpoint of p_mid of P(s)
• Find the vector v^perp perpendicular to P(s)
• The parametric equation of the fold is then:

${\displaystyle F(s)=p_{\mathrm{m}\mathrm{i}\mathrm{d}}+s\cdot {\mathbf{v}}^{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{p}}.}$

Axiom 3

Given two lines l₁ and l₂, there is a fold that places l₁ onto l₂.

This is equivalent to finding a bisector of the angle between l₁ and l₂. Let p₁ and p₂ be any two points on l₁, and let q₁ and q₂ be any two points on l₂. Also, let u and v be the unit direction vectors of l₁ and l₂, respectively; that is:

${\displaystyle \mathbf{u}=(p_2-p_1)/\left|(p_2-p_1)\right|}$

${\displaystyle \mathbf{v}=(q_2-q_1)/\left|(q_2-q_1)\right|.}$

If the two lines are not parallel, their point of intersection is:

${\displaystyle p_{\mathrm{i}\mathrm{n}\mathrm{t}}=p_1+s_{\mathrm{i}\mathrm{n}\mathrm{t}}\cdot \mathbf{u}}$

where

${\displaystyle s_{int}=-\frac{{\mathbf{v}}^{\perp }\cdot (p_1-q_1)}{{\mathbf{v}}^{\perp }\cdot \mathbf{u}}.}$

The direction of one of the bisectors is then:

${\displaystyle \mathbf{w}=\frac{\left|\mathbf{u}\right|\mathbf{v}+\left|\mathbf{v}\right|\mathbf{u}}{\left|\mathbf{u}\right|+\left|\mathbf{v}\right|}.}$

And the parametric equation of the fold is:

${\displaystyle F(s)=p_{\mathrm{i}\mathrm{n}\mathrm{t}}+s\cdot \mathbf{w}.}$

A second bisector also exists, perpendicular to the first and passing through p_int. Folding along this second bisector will also achieve the desired result of placing l₁ onto l₂. It may not be possible to perform one or the other of these folds, depending on the location of the intersection point.

If the two lines are parallel, they have no point of intersection. The fold must be the line midway between l₁ and l₂ and parallel to them.

Axiom 4

Given a point p₁ and a line l₁, there is a unique fold perpendicular to l₁ that passes through point p₁.

This is equivalent to finding a perpendicular to l₁ that passes through p₁. If we find some vector v that is perpendicular to the line l₁, then the parametric equation of the fold is:

${\displaystyle F(s)=p_1+s\cdot \mathbf{v}.}$

Axiom 5

Given two points p₁ and p₂ and a line l₁, there is a fold that places p₁ onto l₁ and passes through p₂.

This axiom is equivalent to finding the intersection of a line with a circle, so it may have 0, 1, or 2 solutions. The line is defined by l₁, and the circle has its center at p₂, and a radius equal to the distance from p₂ to p₁. If the line does not intersect the circle, there are no solutions. If the line is tangent to the circle, there is one solution, and if the line intersects the circle in two places, there are two solutions.

If we know two points on the line, (x₁, y₁) and (x₂, y₂), then the line can be expressed parametrically as:

${\displaystyle x=x_1+s(x_2-x_1)}$

${\displaystyle y=y_1+s(y_2-y_1).}$

Let the circle be defined by its center at p₂=(x_c, y_c), with radius ${\displaystyle r=\left|p_1-p_2\right|}$. Then the circle can be expressed as:

${\displaystyle (x-x_c{)}^2+(y-y_c{)}^2=r^2.}$

In order to determine the points of intersection of the line with the circle, we substitute the x and y components of the equations for the line into the equation for the circle, giving:

${\displaystyle (x_1+s(x_2-x_1)-x_c{)}^2+(y_1+s(y_2-y_1)-y_c{)}^2=r^2.}$

Or, simplified:

${\displaystyle a{s}^2+bs+c=0}$

where:

${\displaystyle a=(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

${\displaystyle b=2(x_2-x_1)(x_1-x_c)+2(y_2-y_1)(y_1-y_c)}$

${\displaystyle c=x_c^2+y_c^2+x_1^2+y_1^2-2(x_c{x}_1+y_c{y}_1)-r^2.}$

Then we simply solve the quadratic equation:

${\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}.}$

If the discriminant b² − 4ac < 0, there are no solutions. The circle does not intersect or touch the line. If the discriminant is equal to 0, then there is a single solution, where the line is tangent to the circle. And if the discriminant is greater than 0, there are two solutions, representing the two points of intersection. Let us call the solutions d₁ and d₂, if they exist. We have 0, 1, or 2 line segments:

${\displaystyle m_1=\overline{p_1{d}_1}}$

${\displaystyle m_2=\overline{p_1{d}_2}.}$

A fold F₁(s) perpendicular to m₁ through its midpoint will place p₁ on the line at location d₁. Similarly, a fold F₂(s) perpendicular to m₂ through its midpoint will place p₁ on the line at location d₂. The application of Axiom 2 easily accomplishes this. The parametric equations of the folds are thus:

${\displaystyle \begin{array}{cc}{F}_1(s)& =p_1+\frac{1}{2}(d_1-p_1)+s(d_1-p_1{)}^{\perp }\\ F_2(s)& =p_1+\frac{1}{2}(d_2-p_1)+s(d_2-p_1{)}^{\perp }.\end{array}}$

Axiom 6

Given two points p₁ and p₂ and two lines l₁ and l₂, there is a fold that places p₁ onto l₁ and p₂ onto l₂.

This axiom is equivalent to finding a line simultaneously tangent to two parabolas, and can be considered equivalent to solving a third-degree equation as there are in general three solutions. The two parabolas have foci at p₁ and p₂, respectively, with directrices defined by l₁ and l₂, respectively.

This fold is called the Beloch fold after Margharita P. Beloch, who in 1936 showed using it that origami can be used to solve general cubic equations.

Axiom 7

Given one point p and two lines l₁ and l₂ that aren't parallel, there is a fold that places p onto l₁ and is perpendicular to l₂.

This axiom was originally discovered by Jacques Justin in 1989 but was overlooked and was rediscovered by Koshiro Hatori in 2002. Robert J. Lang has proven that this list of axioms completes the axioms of origami.

Constructibility

Subsets of the axioms can be used to construct different sets of numbers. The first three can be used with three given points not on a line to do what Alperin calls Thalian constructions.

The first four axioms with two given points define a system weaker than compass and straightedge constructions: every shape that can be folded with those axioms can be constructed with compass and straightedge, but some things can be constructed by compass and straightedge that cannot be folded with those axioms. The numbers that can be constructed are called the origami or pythagorean numbers, if the distance between the two given points is 1 then the constructible points are all of the form ${\displaystyle (\alpha ,\beta )}$ where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are Pythagorean numbers. The Pythagorean numbers are given by the smallest field containing the rational numbers and ${\displaystyle \sqrt{1+{\alpha }^2}}$ whenever ${\displaystyle \alpha }$ is such a number.

Adding the fifth axiom gives the Euclidean numbers, that is the points constructible by compass and straightedge construction.

Adding the neusis axiom 6, all compass-straightedge constructions, and more, can be made. In particular, the constructible regular polygons with these axioms are those with ${\displaystyle 2^a{3}^b\rho \ge 3}$ sides, where ${\displaystyle \rho }$ is a product of distinct Pierpont primes. Compass-straightedge constructions allow only those with ${\displaystyle 2^a\varphi \ge 3}$ sides, where ${\displaystyle \varphi }$ is a product of distinct Fermat primes. (Fermat primes are a subset of Pierpont primes.)

The seventh axiom does not allow construction of further numbers. The seven axioms give all the single-fold constructions that can be done rather than being a minimal set of axioms.

An eighth axiom

The existence of an eighth axiom was claimed by Lucero in 2017, which may be stated as: there is a fold along a given line l₁. The new axiom was found after enumerating all possible incidences between constructible points and lines on a plane. Although it does not create a new line, it is nevertheless needed in actual paper folding when it is required to fold a layer of paper along a line marked on the layer immediately below.