Reading the tetrahedral sky

Geometry before controls

1. Two questions connect the whole Explorer

Imagine three equally spaced rods rising from the vertices of an equilateral triangle to a common point. Extend the rods beyond their original endpoints. A plane cutting all three rods produces a triangle. Tilt or rotate the plane and that triangle changes shape.

The forward question is: given the cone and the cutting plane, which triangle is produced? The inverse question reverses this: given a triangle shape and a cone height, how many different cutting configurations produce that shape? Some triangle shapes cannot occur; others can occur in several ways. A triangle-space atlas organizes these answers into a map.

A third viewpoint fixes an actual triangle on your desk and asks where the cone apex could be. As the cone height parameter varies, possible apex positions trace curves in space. These are the inverse-apex strands. The three tabs therefore examine the same geometry through planes, shapes, and apex positions.

First visit: open the Section explorer, select “Elliptic porism,” and change the rotation φ. Then open Triangle-space atlases and select the regular tetrahedron. Finally inspect Inverse-apex strands for the paper’s scalene triangle.

Paper: §1, “Historical setting and scope,” and §2, pp. 2–3.

2. The symmetric cone and the three parameters

The model uses an equilateral base of circumradius 1 in the horizontal plane and an apex S = (0, 0, h). The base vertices are spaced by 120°. The geometry of the three rays from S is determined by the positive number h. Allowing the base to recede along the rays removes an artificial restriction on the cutting plane; the object being studied is the symmetric trihedral cone.

h — the cone’s shape height

Changes the opening angles between its three rays. Because the base radius is fixed at 1, h is dimensionless. In formulas the paper frequently writes H = h².

α — the cutting-plane inclination

Measures the tilt of a normalized cutting plane. The Explorer uses the plane z = −hμ(x + 1), where μ = tan(α)/h. Changing h while keeping α fixed also changes μ.

φ — rotation around the axis

Changes the orientation of the three equally spaced rays relative to the normalized plane. Angles in the controls are in radians. A full turn is 2π radians; π/3 is 60°.

With θ_i = φ + 2πi/3, a forward intersection requires 1 − μ cos θ_i > 0 for all three rays. These inequalities matter: a root of an algebraic equation can describe the wrong half-ray or a limiting, infinitely distant intersection. The app rejects such configurations when counting physical sections.

Two useful heights are h = 1/√2, where the rays are pairwise perpendicular, and h = √2, where the cone is the regular-tetrahedron case. The limit h → ∞ is the parallel-ray or prism limit. A large finite slider value approximates that limit; it is not infinity.

Paper: §2, Proposition 2.1 and the forward-ray inequalities; §6, special heights and limiting geometries.

3. Why ellipses, parabolas and hyperbolas appear

The circular cone through the three rays intersects the cutting plane in a conic passing through the produced triangle’s vertices: its circumconic. Its type is determined by μ:

Here H in an E/P/H legend means a hyperbolic count, whereas H in a height formula means h². The surrounding text distinguishes them. “Hyperbolic section” describes the conic, not an exotic kind of triangle: the three vertices still form an ordinary Euclidean triangle.

A second conic is tangent to the triangle’s three sides. The circumconic and this inconic are projective images of concentric circles of radii 1 and 1/2. A projective transformation preserves straight lines, incidence and tangency, though it need not preserve lengths or angles.

This explains the Poncelet 3-porism: for fixed h and μ, rotating φ produces a family of triangles whose vertices stay on the same circumconic and whose sides stay tangent to the same inconic. The associated projective return map closes after three steps. In a hyperbolic picture, some projective members are not finite physical sections; the app applies the ray filters rather than displaying them as accepted triangles.

Paper: §3, Theorem 3.2; Figures 1–2 on pp. 5–6. Theorem 3.1 introduces the four collinear distinguished points discussed below.

4. Each point in an atlas represents an entire triangle shape

Two triangles related by translation, rotation, reflection or a common change of scale have the same shape for the inverse problem. An atlas therefore does not place individual triangles in ordinary physical space. It places their shapes in a two-dimensional coordinate system.

The angle-barycentric chart

Write the triangle’s angles as A, B, C, with A + B + C = π. Divide by π to obtain three positive weights that sum to 1. Those weights locate a point in an equilateral reference triangle. In the displayed Cartesian coordinates:

x = (B − A)/π,   y = √3 C/π.

The reference vertices (−1, 0), (1, 0), (0, √3) represent limiting angle triples. Interior points represent nondegenerate triangles. The equilateral shape lies at (0, √3/3). The three medians are isosceles seams, where two angles agree. Lines where one angle is π/2 enclose the central triangle of acute shapes; crossing one gives an obtuse triangle. The outer edges represent degenerate limits.

The complex or normalized similarity chart

Place two vertices at A = (−1, 0), B = (1, 0), and write the third as C = (ξ, η), with η > 0. Then ξ and η determine the shape of the labeled triangle. The equilateral triangle is at (0, √3), not at the barycentric chart’s equilateral location. The real axis η = 0 represents degeneracy. This chart is unbounded; the range control selects a viewing window.

The vertical axis is one isosceles locus. Two radius-2 semicircles supply the other equal-side loci. Right triangles occur on ξ = ±1 or on the unit semicircle ξ² + η² = 1. These curves look different from the straight reference lines in the angle chart because the change of coordinates is nonlinear.

Relabeling A, B, C gives six copies of a generic shape in the labeled chart. This creates the visible symmetry of the atlas. It does not multiply the reported unordered inverse count by six.

Paper: §4, “Triangle-shape spaces and orbits”; Figures 3–4, pp. 11–12.

5. How an inverse count is obtained

For side lengths a, b, c, normalize the squared lengths:

q₁ = a²/(a² + b² + c²), with q₂, q₃ defined cyclically;
p = q₁q₂ + q₂q₃ + q₃q₁,   r = q₁q₂q₃.

The symmetric quantities p and r ignore vertex labels. Eliminating the rotation variable reduces the fixed-height inverse problem to a quartic candidate equation in x = μ². The word “candidate” is essential: solving that equation is only the beginning. A candidate must recover a common rotation parameter c = cos(3φ), satisfy both original invariant equations, and pass the strict forward-ray inequalities.

The accepted configurations are separated into elliptic, parabolic and hyperbolic classes. For example, E2 P0 H1 means two elliptic inverse shapes and one hyperbolic inverse shape at the selected height. It does not mean the point represents three different target triangles, or exactly three labeled spatial apex positions.

In the regular case the paper certifies eight open-cell count labels in a fundamental side-order chamber. The browser uses floating-point recovery and can disagree with exact multiplicities extremely close to seams or repeated roots. Increasing image resolution improves spatial sampling, but does not turn floating-point arithmetic into a proof.

Paper: §5, Theorem 5.1; §6.2, Theorem 6.1 and Table 1.

6. Boundaries explain how the answer changes

Inside an open region away from event curves, the inverse count is locally stable. Crossing a boundary can change it. Three event families are overlaid on the fixed-height atlases:

Gold — the parabolic divider

An inverse branch reaches μ = 1. An elliptic realization may become hyperbolic across this curve. The conic-type counts can change even when their total does not.

Dark — the physical fold

Inverse branches meet at a singularity of the forward shape map. A generic fold can create or remove a pair of inverse realizations. The physical restriction matters; not every algebraic branch belongs in the atlas.

Purple dashed — the compactified escape boundary

A limiting configuration reaches the boundary of the physical domain, with an intersection receding to infinity. An odd change in count is therefore possible; not every count change is a two-branch fold.

The pale dashed isosceles and right-triangle loci are geometric reference lines. They help you recognize the target shape and its symmetry. They are not interchangeable with the three event families. Crossings, contacts and coincidences of event curves explain the height transitions; a coarse colored picture alone may hide those incidences.

Paper: §6.2 and Figures 3–4; §7, height bifurcations; §13, “How to read and color an MPT”.

7. The five classifications answer different questions

Physical E / P / H counts

Fix h and move the target triangle shape. The color records the accepted inverse configurations, split by conic type. This is the principal Map of Possible Triangles (MPT). Use the regular preset to compare with the paper’s certified regular atlas.

Positive parabolic-height count

Now the target triangle is fixed but h may vary. The parabolic condition μ = 1 gives an eliminant Q(H; p, r). P2 or P3 here counts distinct positive roots of that height equation. It is not the same quantity as the P entry in a fixed-height E/P/H label. The paper proves the stated eliminant-root classification, while keeping global physical recovery and sign-cell connectedness as separate certification questions.

Planar-apex endpoint count

This counts the planar limits predicted by the endpoint model: three vertex limits and an interior Fermat–Torricelli limit when all angles are below 120°. At 120° the latter merges with a vertex; beyond it only two relevant vertex limits remain. “Planar” refers to the actual apex altitude z → 0 above the fixed target plane. It does not mean the cone parameter h → 0.

Structural strand phase

The Φ1–Φ5 labels are the port’s numerical grouping of strand behavior, using the 120° threshold, the parabolic eliminant boundary, right-triangle structure, and the regular-star condition. These labels are convenient for exploration; they are not the paper’s thirteen intervals of cone height, and they should not be read as a newly certified global classification.

T9 physical section counts

This fixes h = √3, independently of the general height slider. It studies stars at the ninth transition. The paper gives an exact example with normalized squared sides (1/10, 23/60, 31/60) that has no regular realization but one T9 realization. It shows why increasing height does not make inverse sets nested.

Paper: §8, Proposition 8.1 and Figure 9; §9 and Figure 10; §10, triangle strands and stars; Figure 12 on p. 30.

8. Twelve transitions divide height into thirteen open phases

As h changes, the event curves move through triangle space. Most heights merely deform the picture. At distinguished values, curves cross, touch, meet a symmetry seam or change a relevant endpoint incidence. The paper identifies twelve finite transition heights for this symmetry-marked event geometry.

The selector is labeled T1–T12 and displays h. The phase headings display both H and h. Important landmarks are T1: H = 1/2 (orthogonal rays), T5: H = 2 (regular case), and T9: H = 3. Some remaining values are specified by exact polynomial roots in the paper; their decimal displays are approximations.

“Build 13-phase height atlas” chooses one representative in each open interval between transitions, including the intervals before T1 and after T12. It uses your selected resolution for every panel. These representatives are not the critical slices themselves. A multiple root at a transition may count once, and a branch at a physical boundary can be excluded, so a critical slice need not have the same count as either neighbor.

“Just below” and “Just above” multiply the current h by a small factor. They are exploratory probes, not exact one-sided limits. Very close transition values may call for finer manual input. The final phase sample is a finite height, not the prism limit.

Paper: §7, Theorem 7.8 and the twelve-height table; Figures 5–8 on pp. 20–23; §6.3, Proposition 6.2 for the prism limit.

9. From shape counts to spatial strands and stars

Choose an actual triangle ABC in z = 0 and ask where an apex S can be placed so that the three rays SA, SB, SC belong to a cone of the required symmetric type. For one fixed h there may be several isolated apex positions. Vary h and these positions trace one-dimensional curves: the strands.

The paper’s generic numerical picture has strands associated with the triangle’s vertices. Along them, the conic type may change, and certain invariant conditions select distinguished apex positions, called stars:

• Parabolic stars P: μ = 1; they need not all have the same h.
• Transition stars T_i: the strand meets the cone-height slice h = h_i.
• Orthostar H / T1: the apex with three pairwise perpendicular edge directions; the letter H here names a star, not a hyperbolic count.
• Regular stars T5: apex configurations for the regular cone.

The forward height h is a shape parameter of the cone. It is not the physical z-coordinate of the apex after the produced triangle is mapped onto your fixed triangle. A strand may turn relative to height slices, and the counts at successively numbered transitions need not increase monotonically.

The triangle-plane view projects the spatial geometry and shows the center loci. Drag a vertex to change the target triangle, or use a preset to examine an acute, right, obtuse, isosceles or equilateral case. Symmetric triangles can merge branches or admit continuous special families, so the generic scalene picture should not be applied mechanically to them.

Paper: §10 and Figure 11; §6.1 for the orthogonal case; §9 for planar endpoints.

10. The four distinguished points and their planar traces

Every physical section carries four collinear distinguished points in the cutting plane. Once that triangle is mapped onto your fixed triangle, their paths give four different ways to observe the same changing apex geometry.

C1 — apex shadow

The orthogonal projection of the cone apex onto the section plane. Its locus records the shadow of a spatial strand.

C2 — axis intersection

Where the cone’s symmetry axis meets the section plane. It is generally different from the apex shadow.

C3 — circumconic center

The finite center of an elliptic or hyperbolic circumsection. At a parabolic star this center goes to infinity; the app breaks the curve rather than drawing a false segment across a pole.

C4 — inconic center

The center of the tangent inconic. It remains finite at a parabolic star in the setting proved in the paper.

The paper expresses these maps using normalized reciprocal distances from the apex to A, B and C. Their projected barycentric formulas make it possible to relate the numerical curves to algebraic identities. In particular, the axis-intersection locus lies on a quintic curve; that containment does not imply that every point of the quintic is a physical realization.

At a generic acute vertex, the relevant C1 and C2 branches share a tangent but have different signed curvatures. This is a subtle local phenomenon: an overview plot can make the branches look indistinguishable. A large C3 excursion is also not necessarily a software error—it may be a genuine approach to a projective pole.

Paper: Theorem 3.1; §11, Theorem 11.1 and Proposition 11.2; §12, Theorems 12.1–12.2; Figures 13–15 on pp. 34–36.

11. Four investigations to try

1. Compare the two shape charts. Select the regular tetrahedron, leave both atlases visible, and locate the equilateral shape. Identify the right-triangle and isosceles reference loci. The same inverse classification appears in different coordinates; compare Figures 3–4.
2. Distinguish conic type from total multiplicity. Follow a gold divider away from a dark fold. Compare the E/P/H labels on the two sides. Then examine a fold and an escape boundary. Use §5 and §7 to understand why these events are different.
3. Inspect a transition locally. Select T9, switch to the general physical-count classification, and compare just-below and just-above slices. Use “Equilateral detail” and then double-click near an event-curve meeting point. Each zoom recalculates that small region. Compare the exact critical slice separately.
4. Connect a star to a center pole. In Inverse-apex strands choose the paper’s triangle C = (1/4, 27/20), display P and C3, and compare with C4. The former has parabolic poles; the latter does not. Figures 11 and 13–15 provide the reference views.

For a reproducible experiment, record the classification, h, chart, exact viewport and resolution. “Export data” in the atlas tab saves this metadata and its class legend. The image exports preserve the displayed view.

12. High detail without confusing a picture with a proof

The atlas renderer uses adaptive categorical sampling. It first probes a fine baseline of tiles and refines where classifications differ. For fixed-height maps it additionally refines along the analytical event-curve samples, so a uniform-colored tile is not accepted just because a coarse center missed a nearby curve. It then embeds the colored image beneath separate vector curves, axes and text.

Choose 1024² for detailed interactive work, or 2048² for finer boundaries. Use “Enlarge” to give a chart the full available width. Zooming computes the new coordinates at the selected resolution; it does not enlarge the old bitmap. The equilateral-detail shortcut uses the correct location for each chart.

SVG export keeps curves and lettering as vectors and embeds the sampled color image. It is not a claim that every colored region has an exact vector boundary. PNG 4096 export recalculates the same viewport at 4096² before producing a 4096-by-4096 image. Computation takes longer on a phone or for all thirteen phases; changing tabs or parameters cancels obsolete work.

Even an excellent image cannot certify a region smaller than its sampling scale, a repeated root, or an entire connected component. Floating-point tests also have finite tolerances. The paper separates its exact regular-case and algebraic certificates from computational observations concerning global recovery, planar endpoints and some star classifications. Those distinctions remain in force here.

Paper: §13, “How to read and color an MPT”, and the concluding reproducibility discussion. Source and numerical implementation notes: PORTING.md.

Paper reference map

References above use the currently linked public paper, Plane Sections of Symmetric Tetrahedral Cones: Inverse Fibers, Poncelet 3-Porisms, Phase Transitions, and Triangle Loci, Ismail Hammoudeh, August 2026, public version 1.0.0 (38 pages). Page links refer to that PDF, not to the separately prepared free-software submission revision. Section titles provide a stable fallback if the PDF is later replaced.

Return to the Explorer · Open the complete paper · Free software source