Prime Collapse Solids: A New Class of Geometric Objects for Recursive Structure and Energetic Overload Resolution

Regular and semi-regular polyhedra have long been constrained by the requirement that the sum of face angles meeting at each vertex remains below 360°. In familiar convex polyhedra made of equilateral triangles – the Platonic tetrahedron, octahedron, and icosahedron – only 3, 4, or 5 triangles meet at a vertex, totaling 180°, 240°, or 300° respectively . These sub-360° sums yield positive curvature at each vertex, causing the shape to close up into a convex polyhedron . Even the broader families of Archimedean and Johnson solids, which allow multiple types of faces, obey this convexity constraint, never accumulating a full $360^circ$ around any single corner. For example, none of the 13 Archimedean solids or 92 Johnson solids have more than five triangles at one vertex – in fact, five is the maximum in those classical families, occurring in the regular icosahedron and certain Johnson composites, and it corresponds to $300^circ$ of angle . At six triangles per vertex (each $60^circ$), the angle sum exactly equals $360^circ$; rather than forming a closed solid, this configuration unfurls into a flat tiling – the regular triangular tiling of the Euclidean plane . In such a tiling, often denoted by the Schläfli symbol {3,6}, each point is locally flat and the pattern can extend indefinitely without curvature .

What lies beyond this planar threshold? If one attempts to force more than six equilateral triangles to meet at a point – say 7, 8, or even 11 – the Euclidean rules of polyhedra dramatically break down. The local geometry acquires an angular excess: for instance, 7 triangles contribute a total of $7 imes 60^circ = 420^circ$ around a vertex, overshooting a full circle by $60^circ$. In a conventional setting this is “too many” triangles to fit smoothly around a point in flat or positively curved space. The immediate consequence of such angle overload is geometric frustration: the assembly cannot remain flat or convex, and instead it “collapses” into a non-Euclidean configuration. Parts of the structure must buckle or fold out of the plane to accommodate the extra angle . As described by Segerman and colleagues, a real 3D-printed model with seven triangles meeting per vertex “is not flat, and not a nice regular 3D shape either – it’s a chaotic mess where parts of the structure can be almost flat while other parts fold in on themselves” . Pushing the count higher makes the distortion even more extreme: forcing eight triangles at a vertex “just makes everything even messier” . These qualitative observations hint that when the angular sum exceeds $360^circ$, the structure must develop negative curvature at those vertices (in the sense of saddle-shaped or hyperbolic geometry) to find a new equilibrium .

This paper formally introduces Prime Collapse Solids as a new class of structures that operate in this previously forbidden regime of angular excess. A Prime Collapse Solid (PCS) is defined as a connected 3D structure composed of equilateral triangular faces, where each vertex has a prime number $p$ of faces meeting, and critically, $p$ is chosen such that the total angle at each vertex $Theta = p imes 60^circ$ exceeds $360^circ$. The focus on prime numbers (7, 11, 13, …) for the vertex valence is partly for mathematical elegance and symmetry: primes represent an indivisible count of faces that cannot be factored into smaller symmetric sub-clusters, thus the curvature “frustration” at a prime-valence vertex is in a sense an irreducible phenomenon. (By contrast, a composite number of faces might be conceptually splittable into symmetric groups – for example, 8 faces could be viewed as two sets of 4 – whereas a prime number like 7 or 11 must be accommodated as a whole around the vertex, leading to unique structural arrangements.) Moreover, the primes $p=2,3,5$ correspond to the small cases (degenerate or convex Platonic solids) while $p=7, 11, 13, …$ give the novel collapsed geometries. We deliberately exclude the case $p=6$ since it yields a flat periodic tiling rather than a collapsed solid. Thus, the Prime Collapse category begins with $p=7$ – the smallest prime that produces an angular surplus (420° at each vertex) – and extends to larger primes for potentially even more complex collapses. By focusing on prime counts, we ensure that each vertex in a given structure is identical (a prime-valent homogeneous tessellation), distinguishing these solids as a well-defined family in contrast to arbitrary non-convex deltahedra.

In the remainder of this paper, we develop the theoretical foundations of Prime Collapse Solids and explore their implications. In the Definitions section, we lay out the formal rules and parameters that characterize this class, including the notion of angular excess (negative defect) at a vertex and allowed geometric transformations (folds, concavities, layerings) that resolve the excess. The Mathematical Framework section analyzes the geometry and topology of these structures: we quantify the curvature at a collapsed vertex, discuss the necessity of saddle-like configurations, and derive how such vertices can be assembled into finite or infinite complexes (often requiring higher-genus topology or self-intersecting immersions to exist in Euclidean 3-space). We illustrate the special case of $p=7$ in detail – as it is the gateway to hyperbolic behavior – and examine the further tension that emerges at $p=11$ as a representative larger prime. In Modeling Approaches, we present strategies to construct or simulate Prime Collapse Solids, from physical models (e.g. hinged 3D prints, origami-like folded complexes) to computational tiling methods (mapping hyperbolic tessellations onto surfaces or shells). We show how controlled folding and modular shell layering can transform the theoretical tilings into realizable structures without simply degenerating into chaos. Next, the Applications and implications are discussed: we draw parallels to classical families of polyhedra to emphasize what makes Prime Collapse Solids fundamentally new, and we explore potential uses in materials science (e.g. designing meta-materials with built-in curvature or energy-storing deformation), in architecture (novel saddle-domes and structural forms with negative Gaussian curvature), and in mathematical topology and design algorithms (where these solids intersect with concepts like symbolic tilings, recursive subdivisions, and highly symmetric “Hurwitz” surfaces). Finally, the Conclusion summarizes the significance of Prime Collapse Solids and suggests directions for future research, such as classifying all prime-based collapsible polyhedra or exploring mixed-prime structures, thereby opening a new chapter in geometric exploration.

Before proceeding, we note that the idea of assembling more than six equilateral triangles around a point touches on the rich subject of hyperbolic geometry. In a sense, Prime Collapse Solids can be viewed as finite patches or closed counterparts of regular hyperbolic tessellations. A classical result is that if seven equilateral triangles meet at each vertex, one obtains a tessellation of the hyperbolic plane (often denoted {3,7}) – an infinite 2D surface of constant negative curvature . Mathematicians have long studied such tessellations abstractly, but realizing them materially introduces unique challenges. This work can be seen as bridging the gap: translating the abstract hyperbolic tilings into tangible 3D structures by intelligently allowing folds and concavities. By concentrating on prime-sided vertex figures, we capture the essence of these hyperbolic patterns in a discrete, symmetric manner. In doing so, we also connect to advanced geometric topology: for example, a {3,7} tiling can be “curled up” to form a closed genus-3 surface known as the Klein quartic, though any attempt to physically construct that from flat triangles results in severe warping or self-intersection . Prime Collapse Solids offer a conceptual framework to tackle such constructions in a controlled way, suggesting that what was once “a big mess” might be organized into a new form of structured solid.

Prime Collapse Solid (PCS): Formally, we define a Prime Collapse Solid as a polyhedral structure (not necessarily convex or simply connected) composed entirely of congruent equilateral triangles, such that each vertex of the structure is incident to a fixed prime number $p$ of triangles. We denote such a structure by the symbol PCS$p$ when $p$ is the number of triangles meeting at each vertex. We require $p ge 7$ (since primes below 7 do not produce collapse in the sense of angle surplus). Thus, PCS$7$, PCS11$, PCS13$, etc. are members of this class. Each PCS may be a finite closed polyhedral surface (potentially of higher genus and/or self-intersecting) or an indefinitely expanding mesh (open surface); in practice we often consider finite realizations that approximate the infinite tessellation. Every face in a PCS is an equilateral triangle of side length taken as the unit length for simplicity.

Angular Excess and Defect: At a vertex of any polyhedral surface, define the angular sum $Theta$ as the sum of face angles incident at that vertex. For a triangle-based structure, $Theta = n imes 60^circ$, where $n$ (the vertex valence) is the number of triangles meeting there. In classical convex polyhedra, one always has $Theta < 360^circ$, and the angular defect $D = 360^circ - Theta$ is a positive quantity that measures the local spherical curvature (for example, $D = 180^circ$ for a tetrahedron’s vertices, $120^circ$ for octahedron’s, $60^circ$ for icosahedron’s). In a planar tiling, $Theta = 360^circ$ and $D=0$. In the context of Prime Collapse Solids, we are concerned with the case $Theta > 360^circ$, i.e. a vertex where more than six triangles meet. We define the angular excess $E$ as $E = Theta - 360^circ$ (which is just the negative of the usual defect, hence sometimes called a negative defect). For example, a PCS$7$ has $Theta = 420^circ$ at each vertex, so $E = 60^circ$ at each vertex; PCS11$ has $Theta = 660^circ$, so $E = 300^circ$ at each vertex. This excess angle $E$ cannot be realized in a flat or convex manner around a single point in Euclidean 3-space – a geometric response (folding or curvature) is required to “use up” the extra angle.

Geometric Collapse: We use the term geometric collapse to describe the structural adjustment that occurs when angular excess is present. Rather than the faces neatly spanning a convex angle around the vertex, the vertex “collapses” inward or the faces overlap in a saddle configuration. In practical terms, this means the structure will either (a) fold along some of the edges emanating from that vertex (introducing dihedral angles that deviate from $180^circ$) or (b) form a concave indentation or overlap of material, possibly by layering some triangles above others. The collapse can be thought of as the structure entering a local hyperbolic curvature regime: the vertex acts like a saddle point on the surface. We emphasize that in a PCS, every vertex is such a saddle, since all have $pge7$ faces. This uniformity is a key part of the definition – it echoes the regularity of Platonic solids (where every vertex is identical) but in a dramatically different curvature context.

Allowable Structural Resolutions: A Prime Collapse Solid may resolve its angular excess through several, often combined, mechanisms: • Folded vertices: Some or all of the triangles meeting at a vertex can fold out of the plane of their neighbors. In effect, if one tried to lay $p$ triangles flat around a point, they would overlap; instead, the structure might lift a subset of those triangles above or below the others, creating a fan or crown of triangles that spiral around the vertex. The folds will result in nonzero dihedral angles between adjacent faces. In a controlled fold resolution, these dihedral angles might be chosen consistently so that the excess $E$ is distributed evenly as curvature around the vertex rather than concentrated in a single crinkle. • Structural concavity: The vertex figure (the arrangement of triangles around the vertex) may assume a concave shape in 3D. For instance, some triangles might bend inward (towards the interior of the solid or to one side of a surface) creating a little “dimple” or saddle. This stands in contrast to convex polyhedra, where all vertices point outward. In a concave resolution, the vertex does not lie on the convex hull of its neighboring vertices; instead it might be recessed. The allowable concavity in a PCS is such that the triangles still meet edge-to-edge, but the vertex itself could lie below the plane of any three of its neighbors. • Modular shell layering: In some constructions, it may be convenient to think of a collapsed vertex as comprising multiple layers of triangles that partially overlap or stack. For example, with 7 triangles, one might have a configuration where 6 of them form a roughly closed loop and the 7th one sits on top of (or below) this loop, starting a new layer. The result is that the vertex’s neighborhood is not a single flat disk of faces as in a normal polyhedron, but a multi-layer shell. This concept is particularly useful for higher $p$ (like 11), where one could imagine breaking the $p$ faces into subrings that overlap. All such overlaps must be done in a consistent, non-random manner to still qualify as a coherent PCS – ideally following some symmetric or repetitive rule, rather than arbitrary collision of faces.

We impose that a Prime Collapse Solid, while potentially self-intersecting or concave, should form a single connected surface (possibly with handles or holes) without dangling faces. It should also be composed of rigid triangular facets (we do not allow bending or distorting the faces themselves; all deformation occurs via folding along edges or twisting around vertices). The structures can thus be described combinatorially by the graphs of their vertices, edges, and faces – essentially they are polyhedral complexes – with the key property that the vertex degree (valence) is a fixed prime $p > 6$.

For clarity, we can denote a Prime Collapse Solid by the vertex configuration $3^p$ (using the notation for Archimedean-like vertex configurations, here all faces are 3-gons and $p$ of them meet). Thus in classical terms, Platonic solids gave us $3^3$, $3^4$, $3^5$ as possibilities (tetrahedron, octahedron, icosahedron, respectively) ; the tiling of the plane is $3^6$; and now our interest is in $3^7$, $3^{11}$, $3^{13}$, etc., which are not realizable as convex solids or Euclidean tilings, but will be realized as Prime Collapse Solids. This connects to the theory of regular tessellations: $3^7$ and beyond correspond to regular tessellations of the hyperbolic plane , as will be discussed next.

Curvature and the Hyperbolic Connection: When $Theta < 360^circ$ at a vertex, we have a positive curvature (angular defect) and the local geometry is spherical in nature – this is the case for convex polyhedra, where the defect angles sum to $720^circ$ over all vertices (by Euler's formula) for a closed spherical topology. Conversely, for $Theta > 360^circ$, we have a negative curvature (angular excess) at each vertex, indicative of a hyperbolic geometry. In fact, a uniform tessellation of equilateral triangles with $p$ meeting at each vertex can be associated with a constant curvature surface: for $p=3,4,5$ it fits on a sphere (positive curvature), for $p=6$ on a flat plane (zero curvature), and for $pge7$ it corresponds to a tiling of a hyperbolic plane (negative curvature) . Specifically, $3^7$ (seven triangles at a vertex) is a regular tiling of the hyperbolic plane often denoted by {3,7}, and similarly $3^{11}$ corresponds to a {3,11} hyperbolic tiling, and so on.

The significance of this is that locally, a Prime Collapse Solid with $p$ faces around each vertex will have the geometry of a piece of the {3,$p$} hyperbolic tiling. In hyperbolic geometry, there is more "room" around each point, allowing more than $2pi$ radians of angle around a vertex – intuitively, the surface is saddle-shaped, so triangles can fan out without flattening. However, the full {3,$p$} tiling for $p>6$ cannot be embedded in 3D Euclidean space without distortion; it exists naturally in the hyperbolic plane $H^2$. Prime Collapse Solids essentially borrow the local geometry of $H^2$ (negative Gaussian curvature at each vertex) but then attempt to "close" or realize it in ordinary 3D space by clever folding. They are necessarily non-Euclidean structures in the sense that they cannot be globally flat or convex – some amount of metric distortion or layering is unavoidable. The result is a surface of negative curvature distributed throughout.

Mathematically, one can quantify the curvature per vertex by the angular excess $E = p imes 60^circ - 360^circ$. If one were to treat the PCS as a smooth surface (by smoothing out the sharp folds), each vertex's excess contributes a negative curvature concentrated at that point. For a finite closed surface, the Gauss–Bonnet theorem generalizes Euler's formula: the total curvature (integral of Gaussian curvature) relates to the Euler characteristic $chi$ of the surface. In a polyhedral context, the sum of defects $D$ (for convex) or sum of excesses $E$ (for concave/hyperbolic) across all vertices satisfies $sum_{ ext{vertices}} (360^circ - Theta) = 360^circ chi$. For a usual convex polyhedron (topologically a sphere), $chi=2$ and we get a positive total defect $=720^circ$. For a hyperbolic structure like a PCS, $chi$ will be negative or zero (if it's an open or infinite surface, effectively $chi le 0$). For example, if one attempted to create a closed surface where every vertex has $p=7$ (and thus $E=60^circ$ per vertex), the total curvature would be negative: $sum E = -720^circ (g-1)$ for a genus $g$ surface (since $chi = 2-2g$ for a closed orientable surface of genus $g$). This implies such a surface must have $gge2$ (multiple "holes") to accommodate the negative curvature. Indeed, it is known that a minimum genus for a closed surface admitting a $3^7$ tiling is $g=3$ (a triple torus), realized by the Klein quartic surface which can be tiled by 56 equilateral triangles, 7 around each vertex . The Klein quartic's triangulation has 24 vertices each of degree 7, and it is a highly symmetric case (a Hurwitz surface). Interestingly, the Klein quartic triangulation can be thought of as a finite Prime Collapse Solid of genus 3 (though to embed it in $mathbb{R}^3$ one must allow self-intersections or non-planar embeddings ). Another smaller example: a genus 2 surface (a double torus) can in theory support a $3^7$ tiling with fewer total triangles (our calculations show one possible solution has 12 vertices of degree 7 and 28 faces – though this particular arrangement may not be as symmetric or easy to visualize). The key point is that any closed PCS$_p$ necessarily has Euler characteristic $chi < 0$, meaning it is either a higher-genus polyhedral surface or an open infinite surface. This is a stark departure from Platonic solids, which all live on a sphere ($chi=2$), and even from traditional Archimedean/Johnson solids (also spherical topology). Prime Collapse Solids reside on surfaces that are topologically more complex, aligning with the realm of hyperbolic surfaces in topology .

From a group theory perspective, the condition of $p$ triangles at a vertex relates to a triangle group $Delta(2,3,p)$ – the symmetry group of a {3,$p$} tessellation. For prime $p$, these groups have unique properties; in particular, for $p=7$ we get the (2,3,7) triangle group which is famously the smallest hyperbolic triangle group and is related to the maximal symmetry of certain Riemann surfaces (Hurwitz's $84(g-1)$ theorem) . The symmetry group of the {3,7} tiling, for instance, gives rise to 168 symmetries on the closed genus-3 Klein quartic surface . For $p=11$, the (2,3,11) triangle group is also a hyperbolic group, though not related to a Hurwitz maximal symmetry (since 11 is not of the form $4g+2$ needed for Hurwitz surfaces). Nonetheless, the presence of a prime $p$ often means the only rotational symmetry at a vertex is of order $p$ itself (a $360^circ/p$ rotation), which can impart a high degree of local symmetry to the structure. This underscores that Prime Collapse Solids are not random concave polyhedra, but rather highly regular, albeit non-Euclidean, tilings. In fact, we can think of a PCS as a finite or partial quotient of the infinite {3,$p$} hyperbolic tiling. By "quotient", we mean taking the infinite hyperbolic tiling and identifying edges and faces in such a way as to wrap it up into a finite pattern (possibly with handles or self-intersections). The resulting finite complex inherits a portion of the symmetry and structure of the infinite tiling.

To illustrate the geometric tension for specific primes, consider first the base case $p=7$. With an excess of $60^circ$ at each vertex, a PCS$_7$ must accommodate what is essentially one extra triangle beyond a flat $360^circ$ circle. If one tries to lay out seven triangles around a vertex on a flat surface, the last triangle overlaps the first – there is a $60^circ$ overlap. In a physical model, as soon as you start connecting triangles into a ring, the construction will spontaneously buckle into the third dimension to resolve this. Empirical models and mathematical studies confirm that no matter how you arrange them, a continuous surface of triangles with seven around a vertex cannot remain planar – it will take on a saddle shape with roughly $60^circ$ of total turning angle spread around. One way to picture this is: if you attempt to make a heptagonal "pseudo-pyramid" (analogous to how five triangles form a cap of an icosahedron), the structure will not close but instead form a prismatic saddle. Interestingly, local patches of a 7-triangle tiling can be nearly flat in some areas and highly curved in others . This suggests a degree of flexibility: the exact distribution of curvature isn't rigidly fixed by the combinatorics – the surface can "move" or deform while still maintaining the 7-at-vertex pattern. Indeed, Segerman's hinged 3D print of a {3,7} tiling allowed the model to be flexed in different ways , highlighting a phenomenon unique to hyperbolic tilings: flexibility, as opposed to the rigid fixed shape of a convex polyhedron. Prime Collapse Solids, when constructed with real materials or hinges, may exhibit a similar floppy nature if not stiffened, because the excess angle gives a kind of "slack" that can be redistributed. However, in designing a stable PCS, one might intentionally introduce supporting folds or framework to lock in a particular configuration.

Now consider $p=11$. Here the angular excess is a whopping $300^circ$ per vertex – effectively, each vertex has almost two extra circles worth of angle beyond flatness. This extreme surplus means that naively gluing 11 triangles around a point would result in them winding around nearly twice. One can imagine that a PCS11$, if it exists in a reasonable form, might look like a very tightly ruffled surface, with each vertex perhaps forming a sort of double-layered fan of triangles. The geometric tension at such a vertex is very high: the structure might resolve it by creating a deep concavity, almost a pocket, at each vertex, or by looping some triangles over others. Conceptually, one could arrange 11 triangles such that maybe 5 or 6 form one "ring" and the remaining 5 or 6 form a second ring above it, all meeting at the central vertex – in doing so, the vertex might not be a single point in Euclidean space, but a small complex of points (if layers are separated). In practice, when constructing a PCS11$, one might need to introduce a slight gap or elevation between certain triangles to avoid overlap. This again emphasizes that our definition of PCS does not demand an embedding in $mathbb{R}^3$ as a strictly flat complex – minor self-intersections or layering are permitted as long as the combinatorial incidence (which triangles are considered to meet at a "vertex") is preserved. In this sense, Prime Collapse Solids can also be thought of as abstract polyhedral surfaces (in the graph-theoretic sense) that are realized in 3D with some compromises.

Example – PCS$_7$ (Seven-fold Vertex Solid): To ground these ideas, let us describe a concrete example of a Prime Collapse Solid with $p=7$. One approach to obtaining a finite PCS$_7$ is to start with a known hyperbolic construction. As mentioned, the Klein quartic is a closed surface of genus 3 that admits a $3^7$ tiling with 56 triangles and 24 vertices . It can be described by identifying edges of a $14$-gon in a certain pattern (the combinatorial data is given by a Hurwitz generator). While directly constructing the Klein quartic in $mathbb{R}^3$ results in self-intersecting forms, one polyhedral immersion of it is known as the small cubicuboctahedron (a complex polyhedron which can be visualized with 24 vertices of degree 7). For our purposes, we note that a simpler open construction is also possible: for instance, one can take a portion of the {3,7} tiling shaped like a topological disk and allow it to flare out as a pseudosphere. Mathematician William Thurston demonstrated that you can create a paper model of an infinite hyperbolic plane by successively attaching triangles (this is often done with seven triangles around each vertex, forming a "hyperbolic football" that never closes) . Such a model will inevitably crinkle and overlap at the edges if you try to lay it flat – instead it forms a hand-sized floppy surface with many folds. Segerman's hinged print of a 7-tiling likewise showed that the assembly can "almost" keep expanding but tends to intersect itself and needs to be manipulated in 3D space .

To systematically create a PCS$_7$, one strategy is to use a saddle surface like a hyperboloid as a scaffold. In Huybrechts's work , the idea of placing equilateral triangles on a hyperboloid of one sheet (which has negative curvature except at its waist) was explored to see if seven triangles could meet in a ring on that curved surface. While an exact embedding on a perfect equilateral hyperboloid is tricky (since the curvature isn't constant and the vertices won't all lie perfectly on the surface without adjustment) , this approach suggests that by bending our pieces on a curved form, we can get closer to the ideal {3,7} geometry. In fact, a truncated hyperbolic paraboloid or a periodically curving saddle can host a patch of the {3,7} tiling. For a finite PCS$_7$ structure, one might envision a closed tube or donut shape (genus 2) where the interior negative curvature allows 7-around vertices throughout. Imagine a polyhedral torus that is not flat like a normal torus but has a wavy saddle-like surface – it could potentially be triangulated in a way that every vertex is of degree 7. Such a shape would be a Prime Collapse Solid with $g=2$. If we increase to $g=3$, we reach the Klein quartic example mentioned, which is highly symmetric (each vertex then is indistinguishable under the symmetry group of the surface, which is an interesting property inherited from the tiling's symmetry).

Example – PCS$_{11}$ (Eleven-fold Vertex Solid): Although a concrete physical model for $p=11$ is challenging, we can reason about it theoretically. The {3,11} tiling exists in hyperbolic plane, with each triangle having internal angle $60^circ$ and 11 meeting at each point (so the angular excess per vertex in the hyperbolic plane is $60^circ imes 11 - 360^circ = 300^circ$, meaning the hyperbolic plane's curvature is such that $5pi$ radians (~900°) can fit around a point – hyperbolic geometry allows that because the space is stretching away). Any finite quotient of this tiling will be a surface of some genus $g$ given by a formula similar to earlier. If we attempt a genus $g=6$ for instance, one can solve the Euler characteristic constraints to find a valid triangulation: using the relation $V = rac{3F}{11}$ and $E=rac{3F}{2}$ with Euler $V - E + F = 2-2g$, we find for $g=6$ one solution yields $V=12$ and $F=44$ (interestingly the same 12 vertices as the $p=7$ genus 2 case, but now arranged for $p=11$) – indeed this matches an entry that a 3-11 dual tiling might produce a highly symmetric configuration as well. The numbers suggest a possibility of a genus 6 surface with 12 vertices of degree 11 each, and 44 triangular faces. Whether that can be embedded nicely is another matter; however, in abstract terms it shows that relatively small sized configurations exist. A more brute-force approach to build a PCS11$ is: start attaching triangles one by one around a vertex (like assembling a cone that is over-crimped). By the time you add the 6th or 7th triangle, it will already be curving strongly; adding up to the 11th will probably make it coil around itself. You might then try to continue building outward, hoping to eventually connect edges and close something. An alternate layering could be employed: place, say, 6 triangles in one ring and 5 just above them, all sharing the central vertex. The 6 could form a near-loop (360°) and the remaining 5 would begin a second layer (300° extra). Perhaps that second layer's triangles can bridge to neighboring vertices similarly doing two-layer configurations. While speculative, this hints that PCS11$ might feature a systematic two-level structure at each vertex – a kind of bilayered saddle. In general, as $p$ grows, one expects multiple "levels" of folding might minimize local stress rather than one huge single fold.

Rigidity vs Flexibility: A noteworthy aspect of classical polyhedra is that convex ones are rigid if made of fixed-length struts (Cauchy's rigidity theorem). By contrast, surfaces of negative curvature can often flex. Prime Collapse Solids likely inherit some flexibility if not constrained – for instance, an open mesh of $3^7$ can be deformed in infinitely many ways in 3D . However, if a PCS is constructed as a finite closed surface (with fixed face connectivity), it becomes analogous to a closed hyperbolic surface. Most closed hyperbolic surfaces (of genus $ge 2$) are actually inflexible if realized as a complete geometric structure (they have a fixed hyperbolic metric up to isometry). But when realized in Euclidean 3-space with folds, we must be careful: the folding pattern could introduce mechanisms. In model building, if you use paper or hinged plates, a partially constructed PCS might wiggle – one can press parts and cause others to pop out (like an umbrella turning inside out). To achieve a rigid PCS, one might need to lock certain dihedral angles (perhaps by gluing the structure or adding braces).

In summary, the mathematical framework indicates that Prime Collapse Solids exist at the nexus of combinatorial polyhedron theory and hyperbolic geometry. Each such solid corresponds to a tiling of a surface with negative Euler characteristic by equilateral triangles. The prime number of faces at each vertex gives a uniformity and often high symmetry, while forcing the surface to adopt a saddle geometry. In the next section, we transition from these theoretical considerations to practical modeling approaches for constructing and visualizing Prime Collapse Solids.

Figure 1: A crocheted model of a hyperbolic pseudosphere, illustrating a surface of constant negative curvature (every point is saddle-shaped). Hyperbolic curvature appears when there is an "angle excess," as in the crochet model where adding extra material in each circular row causes the surface to ruffle and fold. This is analogous to forcing more than six equilateral triangles around a vertex: the surplus angle cannot lie flat and instead creates a curved, wavy form. In nature, similar ruffled geometries appear in coral reefs and lettuce leaves due to differential growth (an intrinsic angle surplus in the tissue). Prime Collapse Solids harness this principle in a discrete way – using flat triangles but arranging them with a prime-valence excess at each vertex so that the overall structure takes on a saddle curvature at every vertex.

Designing and constructing Prime Collapse Solids presents unique challenges, as one must reconcile the ideal mathematical tiling with the constraints of physical space. In this section we outline approaches for modeling PCS structures, ranging from theoretical assembly on curved surfaces to concrete physical fabrication techniques.

1. Curved Surface Embedding (Approximate Models): One intuitive way to model a PCS is to start with a continuously curved surface of negative curvature (such as a saddle or a trumpet-shaped pseudosphere) and try to lay out equilateral triangles on it. As noted earlier, a surface like a hyperboloid can support a partial {3,7} tiling . To implement this, one can use differential geometry software or CAD tools: define a target curved surface with Gaussian curvature $K < 0$ (for example, a portion of the $z=cosh(x)$ hyperbolic paraboloid or a catenoid), then project or "drape" a net of equilateral triangles onto it. The advantage is that the surface naturally provides the space for the angles. The drawback is that the triangles may not fit perfectly without distortion, since a generic curved surface might not have all geodesic distances equal. One refinement is to use a numerical relaxation: imagine the triangles as springs or a mesh that can slide on the curved surface, then adjust their positions until all edge lengths equalize and angles match up as closely as possible. Researchers in mathematical visualization have done similar things for representing hyperbolic planes – for instance, by 3D-printing models composed of small panels hinged together . Segerman's 3D printed hyperbolic plane models effectively do this: they print a lattice of polygons (triangles or hexagons) with tiny hinges, which naturally fold into the hyperbolic shape . A PCS can be made by "freezing" such a model into a particular configuration (e.g., by gluing the hinges or using rigid joints instead of flexible ones) once it has taken a desirable shape.

2. Incremental Face-by-Face Construction: Another approach is more combinatorial: build the solid face by face, much like one would build a geodesic dome or an origami model. Start from one seed vertex and attach $p$ triangles around it. This initial assembly will not lie flat – it will form a little cone or saddle. From there, continue adding triangles outward, always maintaining the rule of $p$ meeting at each new vertex. As you add more, you have to decide how to "close" the structure. For an open sheet, you could keep expanding indefinitely (getting a floppy hyperbolic patch). To create a finite closed PCS, you must plan a way for the growth to loop back and meet itself. This often involves some trial and error given the complexity of the pattern. For example, one might assemble a ring of vertices (like a heptagonal ring for PCS$_7$) and then attempt to cap or join edges to form a toroidal structure. In practice, modelers have sometimes used computer search to find how to glue together hyperbolic patches into closed surfaces. For instance, combinatorial searches can rediscover the Klein quartic's gluing pattern by looking for a finite arrangement of 56 triangles where each vertex has degree 7. Tools like Stella (polyhedron net software) or custom scripts can attempt to match edges of a large triangular net in ways that satisfy the vertex valence condition. This is essentially solving a tiling puzzle on a surface of chosen topology.

3. Controlled Folding via Origami/Kirigami: The term "origami" usually refers to folding a continuous sheet, but an analogous concept can be applied to folding a net of triangles. By predefining crease patterns or hinges along certain edges, one can guide how the structure will collapse. For a given $p$, one might design a repeating unit of triangles that fold in a particular way around a vertex. For example, consider PCS$_7$: one could design a module of 7 triangles that meet at a vertex with a specific spiral fold – perhaps each triangle is tilted by about $E/6 = 10^circ$ relative to the next (summing to the 60° excess distributed evenly). This module could then be tiled to form the whole object. Modular origami approaches (like using identical units that assemble into a larger polyhedron) might be adaptable: each unit could enforce the local folding pattern. For instance, one might create a star-shaped seven-triangle unit in paper that has a slight cup shape, and then join many such units edge-to-edge. Somewhat related, there are known "spiral deltahedra" where strips of triangles wind in helical fashion – those are infinite, but one could potentially close a spiral into a loop. A key part of controlled folding is assigning target dihedral angles to edges. In an electronic or simulation environment, one can use optimization algorithms to minimize strain energy such that each vertex's $p$ triangles reach a stable folded configuration. There is precedent in the study of snapology or folded polyhedral nets (though usually for convex shapes); here we push it into concave territory.

4. Layered Assembly (Shells within Shells): The modular shell layering concept from our definitions can be turned into a construction method. Suppose we want to build a PCS$_{11}$. We might attempt a two-layer construction: first create a base layer where (say) 6 triangles meet around a vertex (which by itself would be flat), then add a second layer of 5 triangles that attach around the same vertex but above the first layer. How to attach? One idea: the second layer's triangles might share some of the edges of the first layer's triangles, effectively stacking. This is akin to how one might build polyhedral "honeycombs" or multi-layer structures. If executed perfectly, the vertex in the final structure is not a single point but a small circular ridge where the upper layer sits on the lower layer. However, from a topological graph perspective, that can still be viewed as a single vertex if we consider all those coincident edges meeting. In fabrication terms, one could 3D-print such a structure directly, or use interlocking pieces. Another layering strategy is shell augmentation: start with a known smaller solid (like an icosahedron which has 5 at each vertex) and attempt to attach additional "shells" of triangles around each vertex to raise the count to a prime. For example, take an icosahedron (5 at vertex) and somehow graft 2 extra triangles around each vertex (to make 7). One way is to truncate each vertex slightly, creating a small opening, then insert a little 2-triangle pyramid in that opening. The result won't be regular, but it would yield 7 at what used to be each vertex. This is an ad-hoc example – a systematic approach would involve figuring out how to add patches that increase vertex degree. In general, layering can allow recursive construction: you solve the collapse at one scale and then repeat a similar pattern on a larger scale to close the structure.

5. Digital Simulation and Optimization: Given the complexity of manual construction, computational methods are extremely helpful. One can set up a simulation where each triangle is a rigid plate and edges are hinges with angular springs. By tuning the rest angles of these springs (perhaps wanting a certain angle between plates at equilibrium), one can simulate how the whole mesh will fold up. For a target prime $p$, initially place the triangles as a flat graph (like on a computer, place vertices and connect edges as if it were going to be a planar piece of the tiling). Then introduce a target curvature at each vertex: essentially telling the simulation that each vertex should have a certain total turning (like for 7, each vertex should have a curvature of $D=-60^circ$ i.e., saddle). The system can then use physics (energy minimization) to try to achieve that by moving vertices in 3D. This is analogous to how one might simulate tensegrity or other structural forms. There is existing research on unfolding and folding polyhedral nets reliably – those algorithms often aim to fold a known net back into the target solid. Here, our "target solid" is not initially known – we only know the net and the valence condition. Nonetheless, iterative solvers (like dynamic relaxation or particle-spring systems) can converge to a configuration that satisfies the geometric constraints (within tolerance). The output would be coordinates for each vertex in 3D. If further refinement is needed, these coordinates can be adjusted with CAD software and then fabricated.

6. 3D Printing and Physical Prototyping: Once a digital model of a PCS is obtained, 3D printing is a natural way to bring it to reality. Henry Segerman's approach with the hyperbolic plane was to print the pieces with small hinges so that the model remains flexible . For Prime Collapse Solids, one might print the structure in a fixed configuration (using a rigid plastic or resin) if a stable form is known. Alternatively, printing it in a flexible material or with joints allows experimentation: one could physically push and pull on the model to explore different collapsed states (taking advantage of the flexibility of hyperbolic nets). For example, a PCS$_7$ print could likely be "wrung" into multiple shapes – one might find a symmetric one that is aesthetically pleasing or structurally balanced and then cure or lock the model in that state (perhaps by adding a stiffening spray or attaching a frame). Another tactic is laser-cutting and assembly: print or cut flat equilateral triangles out of paper, cardstock, or plastic, then manually join them with tape or glue in the required pattern. This is essentially making a large net and then attempting to fold it. The builder must follow the pattern carefully to ensure the prime valence condition. During the folding, as expected, it will start forming concave regions. This hands-on method was done in the past for attempts at {3,7} models – for instance, some math enthusiasts have used paper to create partial {3,7} surfaces (often ending up ruffled and hard to connect fully) . With patience, one could complete a finite loop (some have achieved a genus-2 sevenfold tiling with paper by leaving a bit of slack at certain joints). If the physical model is too floppily, adding a minimalist frame – thin wires along certain geodesics or a 3D printed scaffold that clips on – can help it hold shape.

7. Architectural Scale Modeling: If one imagines a large-scale version (like a pavilion or architectural structure using the PCS concept), materials like metal struts and panels or tensegrity cables might be used. One can design a framework where nodes (joints) have $p$ struts radiating (for $p=7$ or 11, etc.), forming the edges around a would-be vertex, and then cover the gaps with triangular panels. The framework must allow the correct angles. This becomes a form-finding problem familiar in tensile structures: one might use tools from structural engineering to determine the equilibrium shape of a net of struts with given connectivity. The result could be a self-supporting saddle surface. In fact, a specific case was reported by Huybrechts : by extending an octahedron-based shape with additional triangles, they achieved a configuration where six vertices have seven triangles meeting, and they noted it as a potentially interesting architecture concept. An architect might deliberately incorporate a pattern of seven-fold vertices in a roof design to create an undulating, concave dome that traditional planar triangulations (like geodesic domes) cannot achieve. We discuss more of these possibilities in the Applications section.

In all these approaches, verification is important: one must confirm that the final model indeed has the intended prime number of faces at each vertex. In a complex model, especially one built by flexible simulation, there is a risk that somewhere the valence could deviate (for instance, two triangles might accidentally align flat effectively forming a larger polygon, or some vertex might split into two if layers separate too much). Careful inspection of the adjacency graph is necessary to ensure the result respects the PCS definition.

The successful modeling of a Prime Collapse Solid often yields a fascinating object: it is neither a classic polyhedron nor a smooth surface, but something in-between – a piecewise-flat surface with intrinsic negative curvature. In the next section, we will look at the properties and potential uses of these objects, now that we have methods to realize them in theory and practice.

The introduction of Prime Collapse Solids as a new geometric class carries implications across several fields – from pure geometry and topology to practical engineering and design. Here we explore various avenues:

A. Extending Classical Geometry: Perhaps the most immediate application is in the realm of geometric research itself. Prime Collapse Solids fill a gap in the taxonomy of polyhedral structures. Historically, mathematicians catalogued the Platonic solids (5 convex regular polyhedra), the Archimedean solids (13 convex semi-regular polyhedra), and the infinite families of prisms, antiprisms, and Johnson solids (92 convex strictly irregular polyhedra) . All of these reside on the convex side of Euler's formula with positive curvature at vertices. On the other end, infinite planar tilings like {3,6} and other Euclidean or hyperbolic tessellations were studied more in tiling theory or abstract polytope theory than as "solids." Prime Collapse Solids bridge these worlds by taking a regular tiling that was previously thought of as purely hyperbolic and giving it a solid-like identity – a discrete structure that one can conceive of as a single object, not infinite plane. In doing so, PCS provide tangible examples of polyhedral surfaces of higher genus, sometimes called Platonic surfaces when they are highly symmetric . For instance, the genus-3 Klein quartic surface we discussed can now be thought of not just as an abstract Riemann surface or algebraic curve, but as a Prime Collapse Solid (if one allows a self-intersecting immersion in $mathbb{R}^3$). This gives a new visualization and didactic tool: students of geometry can physically see and hold a negatively curved polyhedral surface. It also raises new questions for classification: How many distinct Prime Collapse Solids exist for a given prime $p$? Are they all topologically equivalent (same genus) or are there multiple genus solutions? Are there analogues of Archimedean solids where perhaps two different primes or a mix of face types occur (e.g., a solid where some vertices have 7 and some have 11 triangles)? While our definition focused on a uniform prime $p$ throughout, one could relax that to explore semi-uniform cases (maybe as long as every vertex is prime-valent, or maybe exactly two valences allowed). Such explorations would deepen understanding of how curvature can be distributed in polyhedral networks.

B. Materials Engineering – Curvature-Induced Strength and Flexibility: Structures with built-in curvature often exhibit unique mechanical properties. A Prime Collapse Solid's continuous saddle geometry could be advantageous for certain mechanical or architectural purposes. For example, a doubly-curved triangular mesh can be stiffer against certain loads than a flat mesh. In architecture, this is known from thin shell structures: a saddle shape can carry loads efficiently due to its Gaussian curvature. If one constructed a PCS dome (say a roof structure where every node is a 7-fold meeting of beams), it might support itself differently than a typical geodesic dome. The concave regions might help redirect forces and could also create interesting acoustics (focusing or dispersing sound in novel ways). In materials science, one can think of metamaterials or auxetic materials – these are engineered microstructures that have unusual bulk properties (like negative Poisson's ratio, meaning they expand laterally when stretched). A periodic arrangement of triangular units with a certain fold can behave auxetically. Indeed, patterns derived from hyperbolic tilings have been proposed for auxetic materials because the saddle shapes can flex open under tension. A PCS realized at the micro-scale (using perhaps laser-cut micro-trusses or 3D printed lattices) could serve as a unit cell in such metamaterials. Its ability to collapse might allow energy absorption: imagine an impact on a PCS lattice – the initially saddle-shaped vertices could flatten out (absorbing energy as they go from $Theta>360^circ$ towards 360°). After the impact, the structure might spring back, thanks to the stored elastic energy in the bent facets. This energetic behavior (storing and releasing energy via geometric deformation rather than material deformation) is an attractive property for shock absorbers or deployable structures.

Another potential advantage is thermal stability. Some dome designs use curvature to accommodate thermal expansion (the dome can flex a bit without cracking). A Prime Collapse Solid, with its myriad folds, might expand or contract more evenly across its surface by adjusting those folds slightly, possibly avoiding concentrated thermal stress. The fact that $p$ is prime is not directly a factor here, except that prime valence often means more uniform distribution of folds.

C. Symbolic Topology and Mathematics Education: The term "symbolic topology" in our context can be interpreted as the interplay of discrete patterns (like prime numbers) with topological constructs. Prime Collapse Solids offer a very visual symbol: the number of triangles at a vertex is prime, giving each vertex a kind of numeric signature. This could be used pedagogically – for instance, to illustrate the concept of prime numbers in a geometric way. A teacher could show models of 5-around (icosahedron), 7-around (a saddle polyhedron), etc., to tangibly differentiate prime vs composite configurations. On a more advanced level, these structures connect to group theory. The fact that $p$ is prime means the local rotation symmetry is $mathbb{Z}_p$, a cyclic group of prime order, which in many cases can relate to simple groups or projective groups (for example, $mathrm{PSL}(2,7)$ acting on the Klein quartic's 56 triangle tiling , or $mathrm{PSL}(2,11)$ acting on a (2,3,11) triangle orbifold, etc.). This is symbolic in that these solids become concrete realizations of otherwise abstract symmetry concepts. In topology, one could imagine each PCS as a sort of ornamental representation of a particular fundamental group. For example, the fundamental group of a genus $g$ surface with a {3,$p$} tessellation can be related to a triangle group quotient. Prime Collapse Solids thus might serve as mnemonic or visual symbols for those groups or surfaces. A researcher might even use them as physical aids when thinking about phenomena like covering spaces or tessellation of the universal cover (which is the hyperbolic plane tessellation in this case).

D. Recursive Design Systems: One intriguing speculative application is in recursive design or fractal-like architecture. The concept of "collapse generating recursive structures" means that the pattern of collapse at one scale might be repeated at larger scales, creating a self-similar design. For instance, consider a PCS$_7$ where at each vertex the way the triangles fold is somewhat like a small 7-point star concavity. If one looks at the entire shape, one might notice an overall 7-fold symmetry or repeating motif. One could then imagine taking that entire PCS$_7$ solid and using it as a "mega-vertex" in an even larger structure. Because the PCS$_7$ itself has concave regions, perhaps seven of those concavities could meet to form a next-level vertex of a larger construction. In other words, using a PCS as a building block to make a meta-PCS. This is analogous to how one can use an icosahedron to make a geodesic sphere of higher frequency – except here each "node" would itself be a little hyperbolic patch. This kind of recursive assembly could lead to extremely complex but highly hierarchical structures, where each level's prime number pattern influences the next. Why do this? Hierarchical structures are known for multi-functional properties (e.g., in biology bone has micro, meso, macro structure optimizing strength at various scales). A recursive PCS might have, say, shock absorption at the micro level and shape integrity at the macro level, or could be used to create sculptural art that echoes a theme at different sizes.

E. Architecture and Art: As hinted, architects have been intrigued by hyperbolic and negatively curved structures for their aesthetic and structural qualities. Prime Collapse Solids could inspire new architectural forms – for example, a vault or dome composed of triangular panels where each meeting of panels involves 7 panels instead of the usual 3 or 4. This would create a deeply corrugated surface, potentially useful for things like acoustic dampening or diffusing light (the concavities could hold lighting fixtures or acoustic material). Huybrechts's mention of an architectural application suggests that even a partial implementation (like only some vertices have 7, not all) can be interesting. One could design a free-form building skin that predominantly is made of hexagons (flat) but occasionally inserts a heptagon surrounded by 7 triangles (thus creating a negative curvature focal point). Actually, the truncated order-7 tiling (a mix of heptagons and triangles) is sometimes called a hyperbolic soccer ball pattern and has been proposed for architectural tiling because it alternates flat and curved regions.

In the art world, the concept of prime-based collapse might be used symbolically – e.g., an art installation that physically embodies the number 7 or 11 in its structure, perhaps to comment on themes of unity or indivisibility. The visual complexity of these solids, with their recursive folds and spikes, is striking and could be an artistic end in itself. Already, crocheted hyperbolic surfaces have been displayed as art (for example, the Crochet Coral Reef project uses hyperbolic crochet to mimic coral structures). A Prime Collapse Solid made of steel or wood panels could be an eye-catching sculpture in a public space, demonstrating geometric principles while also engaging onlookers with its intricate form.

F. Comparison with Classical Polyhedra – Fundamental Newness: It is worth summarizing what fundamentally sets Prime Collapse Solids apart from the well-known Platonic/Archimedean/Johnson families, beyond the technical definition: • Negative Curvature vs Positive/Zero Curvature: Classical polyhedra (Platonic, Archimedean, Johnson) all have positive curvature at their vertices (except where some Archimedeans include flat tiling-like facets, but still overall they are convex). Prime Collapse Solids have negative curvature at every vertex. This places them in an entirely different geometric category (often requiring a higher-genus topology as discussed). They are, in effect, regular hyperbolic polyhedra in disguise – something that had no counterpart in Euclidean 3D geometry until the advent of modern 3D modeling. In classical geometry books, one could find statements that "fitting 7 equilateral triangles around a point produces a shape that cannot exist in ordinary space" – now with Prime Collapse Solids we qualify that: it cannot exist as a convex polyhedron, but it can exist as a beautiful folded structure in ordinary space . • Vertex-Transitivity and Regularity: Platonic solids are vertex-transitive (all vertices identical). Some Archimedeans are nearly so (they have symmetric orbits of vertices under rotations). Prime Collapse Solids, at least in their ideal mathematical form, are also vertex-transitive in the sense of the tiling. For example, an ideal PCS$_7$ (like the Klein quartic realization) has all vertices identical under its symmetry group. Even a less symmetric PCS will have all vertices of the same degree by definition, which is a kind of regularity. However, unlike Archimedean solids which exist in finite limited count, for PCS there are infinitely many possible ones since one can increase genus or take different quotients of the tiling. Thus, this is a new infinite class of semi-regular structures. They are "regular" in the local sense but require more complex global arrangements. • Structural Behavior: The folds and concavities give Prime Collapse Solids a different structural behavior. For instance, classical polyhedra are rigid frames – a cube or icosahedron is stiff. A PCS might exhibit flexibility or multi-stability (it could perhaps pop from one configuration to a slightly different one if it has some symmetry, analogous to how a saddle can sometimes flip orientation). This means in applications, one could have reconfigurable structures. Imagine a surface that can alternate between two modes: one where the concavities are deeper versus one where they are shallower (somewhat like snapping between two saddle configurations). This could be used in adaptive architecture – panels that change shape to adjust airflow or light. • Thematic and Conceptual Impact: On a more conceptual level, introducing Prime Collapse Solids challenges the notion of what a "solid" is. Traditionally, polyhedra conjure up images of convex dice or perhaps concave star polyhedra (like Kepler-Poinsot which are non-convex but they achieve that by having faces that intersect in a star fashion, not by hyperbolic vertices – in fact, the great icosahedron is a non-convex deltahedron but it has 5 triangles meeting at some vertices with crossing, not >6) . Our solids embrace something that was previously only in the realm of tilings or abstract geometry. This could open minds in mathematics to considering more such hybrids: e.g., could we have a "solid" that locally looks like a piece of 4D geometry or other exotic spaces? In that sense, PCS are a gateway to exploring physical models of non-Euclidean spaces. Already, mathematicians and artists use crochet or paper to model hyperbolic planes; PCS allow modeling a closed hyperbolic surface in a piecewise linear way. This is invaluable for intuition in topology: one can examine loops on the surface, see how geodesics diverge, etc., all on a tangible object.

In conclusion, Prime Collapse Solids carve out a distinctive niche in geometry. They stand as rigorous yet imaginative structures, born from the marriage of number theory (the primeness of the vertex degree), geometry (the interplay of angles and curvature), and topology (the surfaces that accommodate those angles). As we have discussed, their implications and applications are far-reaching: inspiring new questions in pure mathematics, enabling novel designs in engineering and architecture, and enriching our set of metaphors for understanding space and shape (e.g., using them to symbolize the concept of irreducible complexity via primes). They demonstrate that even in a classical field like polyhedral geometry, there are new frontiers to explore when one is willing to venture beyond the traditional limits.

In this work, we introduced Prime Collapse Solids as a fundamentally new class of geometric structures, expanding the landscape of polyhedral forms into the realm of negative curvature and hyperbolic tilings. These solids are defined by a prime number $p$ of equilateral triangles meeting at each vertex, with $p \ge 7$ ensuring that the local angular sum exceeds $360^\circ$ and induces a geometric collapse resolved through folds, concavity, or layering. We developed a formal framework for understanding these structures: mathematically, a PCS$_p$ corresponds to a discrete realization of the {3,$p$} tiling on a surface with Euler characteristic $<0$, often requiring higher-genus topology or self-intersecting immersion to exist in Euclidean 3-space.

Comparative analysis with classical geometric families highlights the novelty of Prime Collapse Solids. Unlike Platonic solids and their convex relatives – which achieve symmetry through positive curvature and are limited to valences of at most 5 for triangular faces – a Prime Collapse Solid achieves a kind of hyperbolic regularity, with identical saddle-shaped vertices and a uniform prime valence beyond the Euclidean limit. In contrast to Archimedean and Johnson solids, which add complexity through multiple face types or lower symmetry but still cling to convexity, our PCS embrace concavity and angle overload as first-class features. The result is a structure that is neither arbitrary nor chaotic, but modularly ordered in its own right – governed by the rigorous rule of prime valence yet free to explore non-Euclidean configuration space.

We illustrated the geometric tension that arises at primes like 7 and 11, where the angular excess per vertex (60° for $p=7$, a hefty 300° for $p=11$) forces creative resolutions. In the $p=7$ case, the structure naturally adopts saddle-like forms, as evidenced by both mathematical models and physical prototypes . We showed that this corresponds to the curvature of a hyperbolic plane patch, and indeed linked the PCS$7$ concept to known surfaces like the Klein quartic (genus 3 with a 7-triangle tessellation) . For $p=11$, we reasoned how multi-layer folding or deep concavity might accommodate the extreme surplus, suggesting that PCS11$ would be an intensely corrugated surface potentially organized in two radial layers of faces. These examples demonstrate how collapse at the vertices can generate recursive and energetic structures: the folding pattern at a single prime vertex can repeat across the surface, creating self-similar motifs, and the overall structure can store energy in its bent facets – energy that might be harnessed if the material is elastic (imagine a deployable structure that "springs" into a saddle shape). The collapse phenomenon can thus be viewed as geometry feeding back into structure: the prime-number rule introduces curvature, which in turn gives the solid new mechanical and aesthetic properties.

Our exploration of modeling approaches – from curved-surface embeddings and origami-like folding to digital simulations and 3D printing – underscores that Prime Collapse Solids, while conceptually abstract, are within reach of fabrication and experimentation. In fact, small-scale prototypes (via paper models or printed hinges) have already captured aspects of the $p=7$ case , and further advances in computational design should enable the construction of more complex cases like $p=11$ in the near future. As these models become reality, we anticipate a rich interplay with applications. We discussed implications for geometry (providing concrete examples of hyperbolic surfaces and prompting new classification questions), for materials engineering (offering designs for auxetic and resilient structures leveraging built-in curvature), for what we termed symbolic topology (connecting prime-based patterns with deep symmetry groups and serving as educational tools to visualize abstract concepts), and for recursive design (enabling hierarchical structures that repeat a theme across scales).

Ultimately, Prime Collapse Solids represent a marriage of rigor and imagination. They are rigorous in that they extend well-defined mathematical tilings and obey precise combinatorial rules, yet they demand imagination to visualize and implement, as they live outside the conventional Euclidean comfort zone. In tone and spirit, they resonate with natural systems – one can see echoes of them in the lattices of certain crystals or the organic patterns of radiolaria skeletons (many of which feature triangulated networks with saddle curvature), and of course in the ubiquitous coral-like forms of hyperbolic crochet . They also connect to artificial systems – from the domes of Buckminster Fuller (which stick to positive curvature) to future "saddle domes" that architects might build by applying these principles .

In closing, the introduction of Prime Collapse Solids expands our conception of what a solid can be. It challenges the longstanding Platonic ideal by showing that regularity and beauty are not confined to convex or flat shapes – they can flourish in negatively curved, recursively folded forms as well. We have only scratched the surface of this new class. Future research may delve into specific instances (e.g., constructing the first explicit PCS$_{11}$ model, or classifying all genus-2 PCS$_7$ variants), explore variations (such as mixing primes or allowing one or two vertices to differ, akin to semi-regular analogues), and optimize practical designs (for example, determining the strongest fold pattern for a given prime lattice under load). The study of Prime Collapse Solids thus opens a vibrant frontier at the intersection of geometry, topology, and design – a frontier where a simple numerical property (primality) catalyzes a collapse of old geometric boundaries and the emergence of new forms.

1. Bennett, J. (2016). The Mathematical Voodoo of Triangles. Popular Mechanics. – Describes how increasing the number of equilateral triangles at a vertex leads to flat tilings and then to chaotic hyperbolic structures (7 triangles "fold in on themselves") .

2. Huybrechts, D. (2012). Polyhedra on an Equilateral Hyperboloid. Bridges Conference Proceedings. – Explores placing polygonal tilings on curved surfaces; notes the possibility and difficulty of seven triangles meeting on a negatively curved surface, and suggests architectural applications of seven-fold symmetry .

3. Mann, K. (2015). DIY Hyperbolic Geometry (Mathcamp Notes). – Provides insight into building hyperbolic planes with paper models; explains that gluing seven triangles at a vertex yields a space of constant negative curvature (the hyperbolic plane) .

4. Wikipedia – Deltahedron. – Background on polyhedra with all-triangle faces; lists classical convex cases and acknowledges infinitely many concave cases .

5. Wikipedia – Order-7 triangular tiling. – Describes the regular tiling {3,7} and its relation to Hurwitz surfaces like the Klein quartic; notes that the Klein quartic can be seen as 56 triangles, 7 at each vertex, on a genus-3 surface .

6. Baez, J. – Klein's Quartic Curve (UCR math webpage). – Presents an accessible explanation of tiling the hyperbolic plane with heptagons or triangles and the construction of the Klein quartic; mentions that trying to build a physical model of the {3,7} tiling results in a "big mess" unless one warps or does "funky maneuvers" (like using pastry!) .

7. Lamb, E. (2012). A Mathematical Yarn: How to Stitch a Hyperbolic Pseudosphere. Scientific American. – Discusses crocheting models of hyperbolic surfaces; notes that adding extra increments (stitches) yields negative curvature causing ruffled forms, similar to what happens with excess triangle angles .

8. Segerman, H. (2016). Visualizing Mathematics with 3D Printing. (Referenced via 3dprintmath.com) – Contains figures of 3D printed hyperbolic tilings; one figure caption notes a seven-triangle-per-vertex tiling printed with hinges "so that it can be arranged in many different ways," indicating the flexibility of such configurations .