Bottema's Zigzag Porism in Projective Geometry

• Bottema's Zigzag Porism is a cyclic closure phenomenon where chains of chords on a circle, intersecting at four collinear points, yield infinitely many inscribed quadrilaterals.
• The formulation employs projective geometry, cross-ratio invariance, and Möbius transformations to show that a single closed zigzag forces the entire mapping to be trivial.
• Recent approaches extend the porism using hyperbolic geometry and integrable-systems theory, connecting it to Darboux’s porism and spatial polygon closure.

Searching arXiv for Bottema’s Zigzag Porism and closely related papers. arxiv_search(query="Bottema zigzag porism cyclic quadrilateral reversion hyperbolic geometry", max_results=10) Bottema’s Zigzag Porism is the closure phenomenon for chains of chords on a fixed circle constrained by four collinear points: if there exists one cyclic quadrilateral inscribed in a circle $\Gamma$ whose side-lines meet a fixed line $\ell$ at prescribed collinear points $P_1,P_2,P_3,P_4$ in order, then there are infinitely many such quadrilaterals on the same circle. In the formulation emphasized by Izmestiev, the porism is equivalent to the statement that a four-step “zigzag” map on $\Gamma$ is the identity as soon as it has one fixed point (Izmestiev, 2014). Kocik recasts the same phenomenon in terms of reversions and Möbius transformations preserving the circle (Kocik, 2014), while recent work places Bottema’s porism, together with Darboux’s porism on folding quadrilaterals, in an Arnold–Liouville integrable framework that also covers the spatial case of two circles in $\mathbb{R}^3$ (Izosimov, 23 Jul 2025).

1. Geometric statement and equivalent formulations

Let $\Gamma$ be a nondegenerate circle, let $\ell$ be a line, and let $P_1,P_2,P_3,P_4\in \ell$ be distinct points not on $\Gamma$. The classical statement is: if there exists an inscribed quadrilateral on $\Gamma$ whose four side-lines meet $\ell$0 consecutively at $\ell$1, then there exist infinitely many such quadrilaterals. Here “sides go through” is understood projectively: the lines extending the edges of the quadrilateral are required to pass through the prescribed points in that order (Izmestiev, 2014).

A standard equivalent formulation uses a chain of chords. Given $\ell$2, set $\ell$3 and define $\ell$4 so that the supporting line of the chord $\ell$5 passes through $\ell$6 for $\ell$7. Writing

$\ell$8

the porism states that if there exists $\ell$9 with $P_1,P_2,P_3,P_4$0, then $P_1,P_2,P_3,P_4$1 on $P_1,P_2,P_3,P_4$2; hence the zigzag closes for every starting point and produces infinitely many inscribed quadrilaterals (Izmestiev, 2014).

Kocik expresses the same construction via reversions. For a point $P_1,P_2,P_3,P_4$3 not on a circle $P_1,P_2,P_3,P_4$4, the reversion through $P_1,P_2,P_3,P_4$5 sends a point $P_1,P_2,P_3,P_4$6 to the unique other point $P_1,P_2,P_3,P_4$7 collinear with $P_1,P_2,P_3,P_4$8 and $P_1,P_2,P_3,P_4$9; this map is an involution. If $\Gamma$0, $\Gamma$1, and $\Gamma$2, then the quadrilateral closes through $\Gamma$3 exactly when

$\Gamma$4

so the four-step composition $\Gamma$5 fixes $\Gamma$6. The porism then becomes: if $\Gamma$7 for one $\Gamma$8, then $\Gamma$9 for all $\mathbb{R}^3$0 (Kocik, 2014).

The 2025 integrable-systems formulation generalizes the term “zigzag” to equilateral polygons alternating between two circles $\mathbb{R}^3$1 and $\mathbb{R}^3$2 in $\mathbb{R}^3$3. In that setting, a zigzag is a polygon $\mathbb{R}^3$4 with $\mathbb{R}^3$5, $\mathbb{R}^3$6, and

$\mathbb{R}^3$7

A closed $\mathbb{R}^3$8-gonal zigzag is one for which $\mathbb{R}^3$9 and $\Gamma$0 (Izosimov, 23 Jul 2025).

2. Projective structure and the cross-ratio formulation

The projective proof is organized around the involution $\Gamma$1 associated with a point $\Gamma$2. For $\Gamma$3, let $\Gamma$4 be the second intersection of $\Gamma$5 with the line through $\Gamma$6 and $\Gamma$7; then $\Gamma$8. This is an involution, extends to a projective transformation of the ambient plane preserving $\Gamma$9, and therefore acts on $\ell$0 as a Möbius transformation. The zigzag map is the composition

$\ell$1

so $\ell$2 is itself a Möbius transformation of $\ell$3 (Izmestiev, 2014).

The key invariant is the cross-ratio. For collinear points $\ell$4 with affine coordinates,

$\ell$5

This quantity is projectively invariant; the paper also recalls the equivalent definitions for four concurrent lines and for four points on a circle (Izmestiev, 2014).

The projective butterfly theorem isolates the exact closure condition. If $\ell$6 meets $\ell$7 in two points $\ell$8, then for a cyclic quadrilateral whose side-lines meet $\ell$9 at $P_1,P_2,P_3,P_4\in \ell$0 in order,

$P_1,P_2,P_3,P_4\in \ell$1

If $P_1,P_2,P_3,P_4\in \ell$2 is tangent to $P_1,P_2,P_3,P_4\in \ell$3 at $P_1,P_2,P_3,P_4\in \ell$4, the corresponding invariant relation is

$P_1,P_2,P_3,P_4\in \ell$5

with signed lengths on $P_1,P_2,P_3,P_4\in \ell$6. If $P_1,P_2,P_3,P_4\in \ell$7 and $P_1,P_2,P_3,P_4\in \ell$8 are disjoint, the equivalent form is

$P_1,P_2,P_3,P_4\in \ell$9

where $\Gamma$0 is the “ideal” point specified by the projectively consistent construction (Izmestiev, 2014).

These three forms play a dual role. They are necessary: closure for one starting point forces the appropriate invariant relation. They are also sufficient: if the relevant relation holds, then the zigzag closes for every starting point on $\Gamma$1. In the secant case the explicit invariant becomes

$\Gamma$2

and this equality is independent of the initial point of the construction (Izmestiev, 2014).

A common projective interpretation is that the porism is not a metric coincidence tied to one quadrilateral; rather, it is the consequence of a line condition on $\Gamma$3 that forces the entire four-step monodromy to be trivial. This suggests why the same mechanism persists under projective changes of coordinates and extends to other nondegenerate conics.

3. Möbius dynamics and the hyperbolic proof

The hyperbolic proof places $\Gamma$4 as the absolute of the Cayley–Klein model. Interior points of $\Gamma$5 are hyperbolic points, chords are hyperbolic geodesics, and the hyperbolic distance between interior points $\Gamma$6 on a chord with endpoints $\Gamma$7 is

$\Gamma$8

Projective transformations preserving $\Gamma$9 are precisely the hyperbolic isometries, and their boundary action is Möbius (Izmestiev, 2014).

In this model, the involution $\Gamma$0 has a direct isometric meaning. If $\Gamma$1 lies inside $\Gamma$2, then $\Gamma$3 is the boundary extension of the hyperbolic half-turn about $\Gamma$4. If $\Gamma$5 lies outside $\Gamma$6, then $\Gamma$7 is the boundary extension of reflection in the polar line $\Gamma$8. Hence every step in the zigzag is simultaneously a projective involution, a Möbius transformation of the boundary, and a hyperbolic isometry of the interior (Izmestiev, 2014).

The classification of Möbius transformations supplies the closure mechanism. Writing a Möbius transformation in projective parameter $\Gamma$9 as

$\ell$00

or equivalently by a matrix $\ell$01 up to scale, orientation-preserving elements are classified by the trace:

• $\ell$02: elliptic, one interior fixed point, no boundary fixed point;
• $\ell$03: parabolic, one boundary fixed point;
• $\ell$04: hyperbolic, two boundary fixed points (Izmestiev, 2014).

Izmestiev’s analysis separates three configurations of $\ell$05 relative to $\ell$06. If $\ell$07 is secant and meets $\ell$08 at $\ell$09, then both $\ell$10 and $\ell$11 are fixed by $\ell$12; if there is any third fixed point $\ell$13, then $\ell$14 must be the identity. If $\ell$15 is tangent at $\ell$16, then $\ell$17 is parabolic with fixed point $\ell$18 or is the identity; any second fixed point on $\ell$19 forces the identity. If $\ell$20 is disjoint from $\ell$21, then $\ell$22 is elliptic with center $\ell$23 or is the identity; an elliptic cannot fix a boundary point, so the existence of any $\ell$24 with $\ell$25 implies $\ell$26 (Izmestiev, 2014).

The hyperbolic argument therefore reaches the same conclusion as the cross-ratio proof: a single closing zigzag implies that the four-step map is globally trivial, and all starting points close.

4. Reversions, matrix calculus, and algebraic closure

Kocik’s proof normalizes the circle to the unit circle

$\ell$27

and identifies points of the plane with complex numbers. The reversion through $\ell$28 is represented by the fractional linear transformation

$\ell$29

or, up to projective scaling, by the matrix

$\ell$30

For $\ell$31, one has $\ell$32, and

$\ell$33

so $\ell$34 are collinear. Reversions are involutions, and their matrices are traceless representatives in $\ell$35 (Kocik, 2014).

The decisive algebraic lemma is that the composition of three reversions through collinear points is again a reversion. If $\ell$36 are collinear, then after multiplying $\ell$37 and normalizing projectively, one obtains a traceless matrix of the same form,

$\ell$38

where

$\ell$39

Geometrically, $\ell$40 lies on the same line as $\ell$41 (Kocik, 2014).

Once this is established, the porism is immediate. If $\ell$42 is the fourth point determined by the triple $\ell$43, then

$\ell$44

Thus the four-step map is the identity on the circle, so if one quadrilateral closes, every starting point produces a quadrilateral with the same four collinear side-intersections (Kocik, 2014).

The same calculus yields an equivalent closure-point formulation: given three fixed collinear points $\ell$45 and any $\ell$46, the intersection $\ell$47 of the line $\ell$48 with $\ell$49 does not depend on $\ell$50. Kocik also records the invariant identity

$\ell$51

for collinear quadruples, and extends the reduction argument to even strings of reversions, giving a porism for cyclic $\ell$52-gons whose sides meet prescribed collinear points (Kocik, 2014).

A notable algebraic example appears when $\ell$53 is the real axis and the fixed points are $\ell$54. Then

$\ell$55

so the fourth point is

$\ell$56

In the paper this is interpreted as the relativistic addition formula for velocities (Kocik, 2014).

5. Integrable-systems interpretation and the relation to Darboux’s porism

The 2025 formulation studies zigzags between two circles $\ell$57 and $\ell$58 in $\ell$59. The zigzag map

$\ell$60

is defined as the composition of two involutions: first replace $\ell$61 by the second point $\ell$62 satisfying $\ell$63, then replace $\ell$64 by the second point $\ell$65 satisfying $\ell$66. By construction, the edge length $\ell$67 is preserved (Izosimov, 23 Jul 2025).

In angular coordinates $\ell$68 on $\ell$69, each involution is anti-symplectic and has the form

$\ell$70

respectively

$\ell$71

for smooth functions determined by the geometry of the circles. Their composition preserves the symplectic form

$\ell$72

and the first integral is

$\ell$73

(Izosimov, 23 Jul 2025).

This leads to a discrete Arnold–Liouville description. On a regular level set $\ell$74, which is generically a circle, there exist action–angle coordinates $\ell$75 in which

$\ell$76

Equivalently, one defines a rotation number $\ell$77 by

$\ell$78

If one orbit on the level $\ell$79 is $\ell$80-periodic, then $\ell$81, so the entire level circle is $\ell$82-periodic and

$\ell$83

on $\ell$84 (Izosimov, 23 Jul 2025).

The resulting theorem is the spatial version attributed there to Bottema and Black: if there exists a closed $\ell$85-gonal zigzag of edge length $\ell$86 between two circles in $\ell$87, then for every pair $\ell$88 with $\ell$89, the zigzag issued from $\ell$90 closes in $\ell$91 steps. In particular, there are infinitely many such closed polygons (Izosimov, 23 Jul 2025).

In the coplanar case, this becomes equivalent to Darboux’s porism on folding quadrilaterals. If $\ell$92 and $\ell$93 lie in the same plane with centers $\ell$94, then the zigzag step is exactly folding the quadrilateral $\ell$95 at $\ell$96 and then at $\ell$97. The quadrilateral moduli space $\ell$98 carries the symplectic form

$\ell$99

the folding map preserves $P_1,P_2,P_3,P_4$00, and the first integral is $P_1,P_2,P_3,P_4$01. The porism mechanism is then identical: periodicity of one point on a regular level set forces periodicity of the whole level set (Izosimov, 23 Jul 2025).

6. Generalizations, edge cases, and historical placement

Several edge cases are treated explicitly in the sources. When $P_1,P_2,P_3,P_4$02 is tangent to $P_1,P_2,P_3,P_4$03 or disjoint from $P_1,P_2,P_3,P_4$04, the porism remains valid and is encoded by the tangent and disjoint variants of the butterfly identity as well as by the hyperbolic classification argument (Izmestiev, 2014). A point at infinity on $P_1,P_2,P_3,P_4$05, corresponding to parallel chords, is described as another instance of the tangent/disjoint dichotomy after a projective change of coordinates (Izmestiev, 2014). Kocik likewise notes that points at infinity can be treated by inversion, and that the line of side-intersections need not intersect the circle (Kocik, 2014).

The treatment of coincident points differs slightly across formulations. In the main projective-hyperbolic statement, the four points on $P_1,P_2,P_3,P_4$06 are distinct and outside $P_1,P_2,P_3,P_4$07 (Izmestiev, 2014). Kocik allows the points to be “not necessarily distinct,” and the reversion formalism includes the degenerate case $P_1,P_2,P_3,P_4$08, for which $P_1,P_2,P_3,P_4$09 for all $P_1,P_2,P_3,P_4$10 (Kocik, 2014). Izmestiev states that if some $P_1,P_2,P_3,P_4$11 coincide, repeated involutions occur, and the porism still holds in the four-point collinear even-chain setting provided a nontrivial closure occurs, although degeneracies can prevent zigzag closure when too many coincide (Izmestiev, 2014).

Both 2014 papers emphasize projective persistence beyond the circle. By projective invariance, the secant-case cross-ratio statement extends from a circle to any nondegenerate conic (Izmestiev, 2014). Kocik develops an algebraic generalization to quadrics by replacing complex numbers with appropriate two-dimensional Clifford algebras: duplex numbers for the hyperbola and dual numbers for the pair of parallel lines, with corresponding reversion formulas and the same composition mechanism (Kocik, 2014).

The porism also sits near broader closure problems. Izmestiev relates the construction to Castillon’s problem: for $P_1,P_2,P_3,P_4$12 points $P_1,P_2,P_3,P_4$13 not on $P_1,P_2,P_3,P_4$14, the chain map $P_1,P_2,P_3,P_4$15 is Möbius, and if it has three fixed points then it is the identity. In the collinear case, if $P_1,P_2,P_3,P_4$16 is odd there is no solution; if $P_1,P_2,P_3,P_4$17 is even and there is one nontrivial solution, then every starting point is a solution (Izmestiev, 2014). Kocik similarly derives a porism for cyclic $P_1,P_2,P_3,P_4$18-gons from repeated reduction of triples of reversions (Kocik, 2014).

Historically, the 2014 projective-hyperbolic treatment attributes the porism to Kocik (2013) and notes that the “zigzag” formulation and the one-closure-implies-all-closures principle are commonly called Bottema’s Zigzag Porism in olympiad literature, even though that paper itself does not cite Bottema by name (Izmestiev, 2014). The integrable-systems paper names Bottema’s original formulation as N. Bottema (1965), records Roger Black’s 1974 spatial generalization, and identifies B. Csikós’s 2000 work as establishing the equivalence between the planar zigzag porism and Darboux’s porism on folding quadrilaterals (Izosimov, 23 Jul 2025). Taken together, these sources place Bottema’s Zigzag Porism at the intersection of projective geometry, Möbius dynamics, hyperbolic geometry, and discrete integrable systems.