For grown-ups · the house through the looking glass

The House, exactly.

The child’s house is two mathematical objects and five theorems. This page states them, walks the eleven rooms through them in order, and draws every figure from exact arithmetic (integers, rationals and quadratic surds) ported from the Looking Glass and checked against independent oracles. It is a rigorous structural example. It is not a claim about what the world is made of.

▶ Play the tour: the storyteller reads this page, with the figures in motion ✦ Kaleidoscope tour

What the child sees, and what it is

In the house	In the mathematics
the gold dot	a point n of the base S²
the hidden circle, the rail	the fibre π⁻¹(n), a circle
the bead	a point of the fibre: a phase e^{iγ}
the dot cannot tell the whole story	π has no global section
two rings, linked, not joined	two fibres: disjoint, linking number 1
the flat picture	a projection to R², where a false crossing appears
coming home turned	holonomy of a connection around a loop
the gate sign and the smaller signs	constraint sets ordered by inclusion, S ≤ P
keeping what both allow	the meet, P ∧ S
a wise house knows where its map ends	no universal chart: the atlas stays open

Object I

The bundle: the Hopf map

π(z₁, z₂) = (2 z₁ z̄₂, |z₁|² − |z₂|²), with |z₁|² + |z₂|² = 1

π sends the 3-sphere S³ ⊂ C² onto the 2-sphere S². Its fibres are circles: over every point n of S² sits a whole circle of (z₁, z₂), turned by the phase e^{iγ}. Any two fibres are linked exactly once. And π admits no global section: there is no continuous choice of one point on every fibre. Its first Chern number is

∫_S² c₁ = −1

The sign follows the kernel’s orientation, in which the y axis is the opposite of the usual Bloch one; the fact that the number is not zero is what forbids a section. The figures use the exact geometry of the Looking Glass: every centre, radius and crossing is a reduced rational, not a float.

The fibre over n, seen through stereographic projection from the pole (0, 0, 0, 1) of S³: an exact circle with centre (0, 3/4, 0) and radius 5/4. The four marked points are the exact images of the phases t = tan(γ/2) ∈ {0, 1, ∞, −1}. Move round the circle and n does not move.

The fibres over two distinct points of S², mirror images in the plane x = 0. The linking number is certified exactly: 2 signed crossings in a declared diagram halve to 1; the inscribed rational polygons (32 vertices each) are further apart (distance² = 207936/212761) than the sum of their sagitta bounds, so straightening the arcs cannot change the count. Their separation is at least 0.929 in chart units. Linked, not joined.

Object II

The lattice: constraint sets and their meet

μ(P, S) = P ∧ S; a child S under a parent P is admissible only if S ≤ P

The rules of the house form a partially ordered set of constraint sets, ordered by inclusion. Two rules combine by their meet: keep what both allow. A child rule may narrow its parent and may never widen it: loosening, S ≰ P, is not an element of the house at all. This is a meet-semilattice, and it needs no top element that contains everything; rules compose by compatible overlap.

Constraint sets under the gate P, ordered by inclusion. A child S is admissible only if S ≤ P; two rules combine by their meet, S ∧ T, keeping what both allow. Loosening, S ≰ P, is not an element of the house at all. The top is P, not the atlas: nothing here contains everything.

Five theorems carry the weight

G1 · No global section

There is no continuous s: S² → S³ with π ∘ s = id. Locally, over any small patch, sections exist in abundance; globally, none. The visible point never determines the hidden position, and no rule of thumb could make it do so everywhere at once. Walked in Foundation A and Room 6, the bead.

G2 · Fibres are linked, not joined

Any two distinct fibres form a Hopf link: they are disjoint, and their linking number is 1. Neither is the other; neither can be pulled free. A flat picture of them shows crossings that are not there in the room. The certificate below is exact: 2 signed crossings in a declared view, halved to 1; inscribed polygons of 32 vertices further apart than the sum of their sagitta bounds. Walked in Room 7, the link.

G3 · Three connections on one patch give three numbers

On the same sphere, around loops on the same patch, three different connections return three different numbers, and they must never be printed for one another:

π/2: the Levi-Civita turn of a tangent arrow carried around the octant, equal to the octant’s area (Gauss–Bonnet; vertex angles π/2, π/2, π/2).
−π/4: the curvature flux of the kernel’s Hopf connection through the same octant, F = −½ sin θ dθ ∧ dφ; the lift returns with phase π/4.
π: the phase of the horizontal lift carried once around the equator. The lift comes home multiplied by e^{iπ} = −1: the same place, the opposite sign.

Walked in Room 10, the turn.

The loop e₁ → e₂ → e₃ → e₁ is three quarter-arcs of great circles. Carrying the arrow by Levi-Civita transport (never turning it on purpose), it comes home rotated by π/2: exactly the octant’s area, as Gauss–Bonnet says (vertex angles π/2, π/2, π/2; excess π/2). The same loop lifted horizontally to S³ returns with phase π/4 (Bargmann invariant (1/4, -1/4)), while the kernel’s curvature flux through the octant is −π/4. Same loop, three connections, three different numbers.

G4 · Compactness in the E-series is a determinant

Extend the E₈ diagram one node at a time along its long arm and take the determinant of the Cartan matrix: E₈ gives 1, E₉ gives 0, E₁₀ gives -1, E₁₁ gives -2. Positive means finite and closed: a room you can count to the end. Zero is the threshold: affine, with a null direction δ. Negative means hyperbolic (E₁₀) and then merely Lorentzian (E₁₁): infinite, and never the whole. Walked in the closed room (E₈), the hinge (E₉), the count (E₁₀) and the door (E₁₁).

E₈ det 1, signature (8, 0): 240 roots, finite: the closed room. Room 2 · The Closed Room.E₉ det 0, signature (8, 0, 1 null): affine E₈; null vector δ = (1, 2, 3, 4, 6, 5, 4, 3, 2); infinitely many roots, all within the light cone's edge. Room 5 · The Hinge.E₁₀ det -1, signature (9, 1): hyperbolic, root lattice II₉,₁, Weyl vector ρ² = -1240; 75 real roots up to height 8 (10, 9, 9, 9, 9, 9, 10, 10 by height). Room 9 · The Count.E₁₁ det -2, signature (10, 1): Lorentzian, not hyperbolic; 61 real roots up to height 6 (11, 10, 10, 10, 10, 10). Room 11 · The Door.Each diagram is the last with one more node on its long arm (Kac, Table Aff 1; Damour–Henneaux–Nicolai 2003, §4). The determinant of the Cartan matrix crosses zero exactly at E₉: positive and finite before, null at the hinge, negative and infinite after.

G5 · The meet-semilattice

Children narrow and never loosen: S admissible under P only when S ≤ P; composition is the meet. And no room is the atlas: no chart, level, person, institution or system is silently promoted into the map of everything. The house’s last room contains an open door for exactly this reason. Walked in Foundation B, the rule garden, and Room 11, the door.

The eleven rooms walk the theorems in order

Room	Walks	The exact fact
1 Two	the two objects are two	A bundle and a lattice: distinct kinds, never identified.
2 The Closed Room	E₈ (G4)	det 1; 240 roots; finite and complete, and one room of eleven.
3 Same Dots, New Loops	S⁷ → S⁴	the same construction over the quaternions: same points, new fibres.
4 Two Lengths	G₂ (12 roots)	12 roots, six short and six long; squared lengths in ratio 3.
5 The Hinge	E₉ (G4)	det 0: the threshold where finite stops and infinite begins.
6 The Bead	G1	no global section: the fibre over n is a whole circle.
7 The Link	G2	two fibres, linking number 1, disjoint; the flat picture shows 2 false crossings.
8 The Bounce	the cosmological billiard	each bounce is a reflection in a curvature wall (Damour–Henneaux–Nicolai).
9 The Count	E₁₀ (G4)	det -1: hyperbolic; the count of real roots by height is finite in every window and never ends.
10 The Turn	G3	π/2, −π/4, π: three connections, three numbers.
11 The Door	E₁₁ (G4) with G5	det -2, Lorentzian, not hyperbolic; and no room is the atlas.

The three pictures the child touches most, bead, link and turn, are G1, G2 and G3. The door is G4 at E₁₁ together with G5.

Room 3 · S⁷ → S⁴, the quaternionic Hopf map

π(q₁, q₂) = (2 q₁ q̄₂, |q₁|² − |q₂|²), quaternions with |q₁|² + |q₂|² = 1

The same pieces, the same recipe, with quaternions in place of complex numbers: the fibres are now 3-spheres and the base is S⁴. It is exactly rational on lattice points: the pair q₁ = ½(1, 1, 1, 0), q₂ = ½(1, 0, 0, 0) is sent to (1/2, 1/2, 1/2, 0, 1/2), a unit vector. Same dots, new loops.

The G₂ root system in the plane x₁ + x₂ + x₃ = 0 (Humphreys §12.1, Bourbaki Plate IX): short roots eᵢ − eⱼ, long roots ±(2eᵢ − eⱼ − eₖ), squared lengths in ratio 3. Cartan matrix [[2, −1], [−3, 2]], determinant 1, Coxeter number 6, Weyl group of order 12. The picture does not show octonions. Twelve roots: six short and six long, not twelve and twelve.

Kasner exponents p(u) lie on the circle Σp = 1, Σp² = 1. Each BKL epoch change is a bounce: reflection in a curvature wall, drawn in the Klein model as the second meet of the circle with the line from the wall’s pole (Misner 1969; Chitre 1972; Heinzle–Uggla 2009). The wall answers only at the edge; the orbit is its own. The cosmological billiard is Damour–Henneaux–Nicolai (2003). This is a map of Kasner epochs, not a solution of Einstein’s equations, and the E₁₀ chamber, nine-dimensional, is not drawn.

Two corrections stand, and one addition

G₂ has 12 roots: 6 short and 6 long, not twelve plus twelve. Computed above from Bourbaki’s simple roots; the root set is closed under its own reflections and the Weyl group has order 12.
The cosmological billiard is Damour–Henneaux–Nicolai (Class. Quantum Grav. 20, 2003). The bounces drawn in the count’s neighbour, the bounce room, are theirs: reflections in curvature walls of the E₁₀ Weyl chamber, shown here in the Kasner plane.
Addition. The E₁₀ Weyl chamber has finite volume but a single cusp, and that cusp is where the E₉ subdiagram sits: the null direction δ = (1, 2, 3, 4, 6, 5, 4, 3, 2) of the affine E₈. In the house’s order, the hinge (E₉) is the vanishing point of the count (E₁₀): the one place where the finite chamber reaches the edge of the infinite.

Provenance and honesty

The exact modules (`exact`, `quadratic-surd`, `root-system`, `e-series`, `e-series-windows`, `kac-moody`, `g2`, `hopf-s3-exact`, `hopf-cuts`, `s2-transport`, `kasner`, `kasner-billiard`) are ported unchanged, save for a rational shim, from the Looking Glass research code, where each is written from the standard sources it cites (Bourbaki, Humphreys, Kac, Hopf, Lyons, do Carmo, Bargmann, Damour–Henneaux–Nicolai) and checked against independent Python-stdlib oracles. Those oracle fixtures ship with this site and its tests run against them: determinants 1, 0, −1, −2; the 240 roots of E₈; the window counts of E₁₀ and E₁₁; G₂’s twelve roots; the bead’s circle; the link’s certificate; the octant’s three numbers.

What the house claims: a visible point does not determine its fibre; there is no global section; local charts work where one universal chart does not; distinct fibres link without meeting; a lower-dimensional picture can show a false crossing; transport around a loop returns with changed orientation; constraints narrow without inventing permission; composition happens by compatible overlap; nothing is silently promoted into the whole atlas; the structure stays open. What the house does not claim: that reality is a Hopf fibration. The fibration is the rigorous example; the pattern is the lesson.

Back to the house, or hear it told in the story.