A method and apparatus for recursively generating hourglass-shaped and chain-like structures

Abstract

The application provides a method and device for recursively generating hourglass-shaped and chain-shaped structures, which solves the problem of recursive structure demonstration, and the elastic body is made of flexible material and has a closed ring or a spherical body as its inherent stable configuration; the force applying structure is fixed on or formed on the elastic body and is used for applying opposite torque by a user to form a narrow neck. The application has the advantages of stable results and good demonstration effect.

Description

Background Technology

[0001] In the fields of philosophical education, demonstration of scientific principles, cultural interpretation, and interactive experiential installations, there is a persistent need for intuitive and dynamic interpretations of the origin of the universe, structural evolution, and the essence of life. The transformation logic of existing transforming toys or devices is either too simplistic and random, lacking profound meaning, or too complex and mechanical, unrelated to principles of natural philosophy. In particular, there is a lack of a simple, intuitive, and repeatable single physical operation that corresponds to the philosophical principle of differentiation and, through recursive application, logically and necessarily generates complex forms symbolizing the structure of life's information. This makes it impossible for current technology to provide a credible and operable physical demonstration of the interdisciplinary idea of ​​the isomorphism between the universe and life.

[0002] To address the shortcomings of existing technologies, people have conducted long-term explorations and proposed various solutions. For example, Chinese patent literature discloses a spiral-wound double hourglass [200820112580.3], which consists of two hourglasses. Two spiral hourglass steel molds of the same size, thickness, and degree of twist are made. Flower patterns and curved patterns are engraved on the walls of the steel molds to create flower-shaped cylindrical spiral hourglasses and curved cylindrical spiral hourglasses. Then, a spiral-wound double hourglass steel mold is made, and the flower-shaped cylindrical spiral hourglasses are embedded in the spiral-wound double hourglass steel mold. Then, glass solution is blown into this steel mold to create curved cylindrical spiral hourglasses. At this time, the curved cylindrical spiral hourglasses are tightly wound inside the flower-shaped cylindrical spiral hourglasses, forming a spiral-wound double hourglass. Different colored sand is filled in each hourglass, and the spiral-wound double hourglass is placed in a frame.

[0003] The above solution has solved the problem of demonstrating the spiral hourglass structure to some extent, but it still has many shortcomings, such as the inability to demonstrate deformation. Summary of the Invention

[0004] The purpose of this invention is to address the above-mentioned problems by providing a recursive method for generating hourglass and chain-shaped deformation structures that is well-designed and effective for teaching demonstrations.

[0005] Another objective of this invention is to address the aforementioned problems by providing a deformation device for recursively generating hourglass and chain-shaped structures with stable deformation structures.

[0006] To achieve the above objectives, the present invention employs the following technical solution: a method for recursively generating hourglass-shaped and chain-like structures, comprising the following steps: S1: Provide a flexible body, which presents a first closed shape in its natural state; where the natural state refers to the minimum potential energy configuration of the flexible body without external constraints and stress. This configuration is determined by the material elasticity, geometry and preset memory characteristics. The first closed shape is topologically equivalent to a simple closed curve or surface, providing a clear initial topological boundary for subsequent deformation. S2: Determine two initial force application points on the flexible body; S3: Applying a pair of opposite torques to two initial force application points drives the flexible body to deform from a first closed state into a first-stage hourglass shape. This first-stage hourglass shape has a narrow neck in the middle and two bulges at the ends. The opposite torques refer to a pair of torques around the approximate central axis of symmetry of the flexible body. This torque causes the body to undergo complex nonlinear elastic deformation. The waist region becomes unstable under the combined stress of compression and tension, resulting in necking and forming a stable narrow neck. The material at both ends expands outwards after stress redistribution, forming bulges. This process simulates the symmetry breaking and morphological bifurcation of a continuum under specific boundary conditions. S4: Take any one of the enlarged parts in the first-level hourglass shape as the new action area, and apply a new round of torque in the opposite direction to it, so that it recursively deforms into a smaller-scale second-level hourglass shape. As a core step, the selected enlarged part has become a relatively independent substructure on a macroscopic level, but its material still maintains continuity and elasticity. The secondary torque applied to it will repeat the mechanical process of step S3 in the local area of ​​the substructure, inducing secondary necking and chamber differentiation. This reflects the self-similarity and scale invariance of the morphology generation logic. S5: By repeating the above recursive steps, a chain structure is formed by connecting multiple hourglass shapes sequentially through their narrow necks.

[0007] The repetition of recursive operations results in the generation of a series of topologically isomorphic but scale-decreasing hourglass units. These units are connected in series through a shared narrow neck, which is the bulge of the previous level that is then narrowed, forming a hierarchical chain topology in structure; the narrow neck is both a connector and the geometric starting point of the recursive operation.

[0008] In the aforementioned recursive method for generating hourglass and chain-like deformation structures, the first closed form is a two-dimensional closed loop, with the hourglass shape being a planar figure-eight; or the first closed form is a three-dimensional spheroid, with the hourglass shape being a three-dimensional hourglass. The deformation of the two-dimensional loop body is mainly governed by in-plane bending stiffness, and its figure-eight shape is a stable configuration of a planar curve under a specific torque. The deformation of the three-dimensional spheroid body involves complex surface strain, and its three-dimensional hourglass shape is one of the axisymmetric buckling modes of the shell under biaxial stress. Although the two have different dimensions, they share the generative kernel of reverse torque-induced necking.

[0009] In the aforementioned recursive method for generating hourglass and chain-like structures, when the first closed form is a three-dimensional sphere, it contains a flowable medium. During deformation, the flowable medium flows between the bulge and the narrow neck. The introduction of the flowable medium enhances the system's dynamic performance. During deformation, the gravity and pressure distribution of the medium are coupled in real time with the changes in the cavity shape: when the narrow neck forms, the medium passes through the narrow channel driven by the pressure difference, visually demonstrating the communicating vessel effect and changes in flow resistance; the flow trajectory and final distribution of the medium also become visual indicators characterizing the internal stress field and structural stability.

[0010] In the above-mentioned recursive method for generating hourglass and chain-like structures, the chain-like structure formed after at least one recursive deformation presents a double-helix chain-like spatial pattern on a two-dimensional projection.

[0011] When two or more recursive operations are performed, causing the narrow neck axes of adjacent hourglass units to twist at a certain angle—an angle naturally determined by the relative orientation of the recursive force application points—the resulting chain structure, projected onto a two-dimensional plane, will exhibit an interlaced visual effect in terms of the outlines of the bulging portions of its continuous units. This pattern is geometrically highly similar to a simplified projection of a double helix chain. This phenomenon is not intentionally designed, but rather a geometrically inevitable result of the recursive twisting operation from a specific perspective.

[0012] An apparatus for recursively generating hourglass and chain-like structures, comprising: The elastic body is made of flexible material and its inherent stable configuration is a closed loop or a spherical shape; The material of the elastic matrix must possess high resilience, fatigue resistance, and moderate yield strength to ensure it can withstand multiple recursive deformations and reliably recover. Its inherent stable configuration can be achieved through pre-setting during material forming, such as the austenitic phase shape of shape memory alloys; or through the balance between the material's own stiffness and geometry, such as a rubber ring of uniform thickness.

[0013] A force-applying structure, fixed to or formed on an elastic body, is used to allow the user to apply torque in the opposite direction to form a narrow neck.

[0014] The core function of the force-applying structure is to efficiently and controllably transmit the force and torque from the user's hand to specific areas of the elastic body. Its design must consider ergonomics such as anti-slip and comfortable grip, as well as mechanical coupling such as preventing slippage and stress concentration, to ensure that the operating intention can be accurately converted into torque that drives the deformation of the body.

[0015] In the aforementioned recursive deformation device for generating hourglass and chain-like structures, the elastic body is a ring structure made of shape memory alloy or polymer with shape memory properties. Shape memory alloys exhibit both hyperelasticity and shape memory effects, and their stress-strain curves show a distinct plateau region. This allows the ring structure to smoothly and with a large stroke transition from one stable shape to another under torque, with reversible and repeatable deformation. Shape memory polymers, on the other hand, can provide even greater deformation and programmable recovery temperatures.

[0016] In the aforementioned recursive deformation device for generating hourglass and chain-like structures, the elastic body is a sealed capsule made of a transparent, flexible material, filled with a flowing substance in the form of particles or liquid. Transparent, flexible materials such as silicone and TPU allow direct observation of the internal flow process. The sealed capsule must possess good airtightness and puncture resistance. The choice of the flowing substance—sand particles of different colors, densities, and particle sizes, or immiscible liquids—can create rich visual and dynamic effects, transforming the abstract form generation process into an intuitive and traceable flow of matter and changes in interfaces.

[0017] In the aforementioned recursive deformation device for generating hourglass and chain-like structures, the force-applying structure consists of two valves positioned at opposite poles of the sealed bladder. These valves can be used to inflate the sealed bladder, or, after the sealed bladder deforms into a funnel, to act as an automatically locking channel through which a thin, needle-like steel wire is inserted from the outside into the hourglass. This wire moderately enlarges the narrow neck that has closed due to tightening, facilitating the passage of sand. After enlarging the hole, the steel wire can be removed from the valves. Because the sealed bladder is made of a material with memory function, the hole will not close after the steel wire is removed. Furthermore, because the sealed bladder has a flexible, self-locking structure, the hourglass will not leak internal gas after the steel wire is removed.

[0018] In the aforementioned recursive deformation device for generating hourglass and chain-like structures, the elastic body is composed of multiple annular or spherical sub-units connected in series by movable connectors, which allow relative rotation between adjacent sub-units. This achieves a modular and discretized scheme for recursive deformation. Each sub-unit is an independent elastic deformable body. Movable connectors such as miniature universal joints and elastic band loops transmit torque while constraining the relative displacement degrees of freedom between sub-units, making the overall deformation more orderly and controllable. Users can apply torque to any specified sub-unit to achieve precise, path-selectable recursive deformation demonstrations.

[0019] In the aforementioned recursive generation of hourglass and chain-like deformation devices, the force-applying structure includes two symmetrically arranged grips on the elastic body, or specific application areas marked on the surface of the elastic body by color, texture, or physical recesses. These grips can be partially covered soft rubber, raised lugs, or embedded hard sheets, designed to provide clear grip feedback. Marking with color, texture, or recesses represents a more integrated guiding design, visually and tactilely integrating the force application point with the body's shape, guiding the user to spontaneously apply force in the correct location without the need for additional parts.

[0020] Compared with existing technologies, the advantages of this invention are: 1. By actively applying reverse torque, the device is transformed from its initial symmetrical state into a first-order bipartite structure, which reproduces the basic process of symmetry breaking and central constraint generation from a unified body. This dynamic generation process transforms the abstract concept of structural origin into an intuitive and operable physical experience, solving the key defect that result-oriented teaching aids cannot provide process cognition. 2. The device reveals and utilizes a common morphological generation logic, namely, that the reverse torque acting on a flexible closed body can induce necking and chamber differentiation. This logic is not only applicable to the generation of primary structures, but can also be recursively applied to the generation of new chambers, thereby deriving multi-level, chain-like composite structures. At the physical entity level, the generation of simple dichotomous structures and complex chain-like structures is attributed to the same mechanical principle, providing a unified and experimental model for understanding the emergence of structures at different scales and domains. 3. The same reverse torque operation can be performed again on the formed structural unit, thereby generating a smaller-scale secondary structure within the unit. This makes the complex chain structure not generated all at once, but grows step by step through rule repetition. The final double helix chain projection pattern is the natural geometric result of this recursive operation on the two-dimensional plane. It transforms the mathematical concepts of recursion, iteration and fractals into a physical program that can be executed by hand. 4. The interaction is simple and requires no learning cost. It directly corresponds to several deep principles such as torque deformation in mechanics of materials, continuous transformation in topology, and structural emergence in systems science. It is suitable for different cognitive stages from intuitive cognition to abstract thinking. 5. It can be achieved through a variety of materials and structural forms. Diverse product forms can be derived based on different costs, durability, and demonstration needs, making it widely applicable in fields such as basic science education, engineering technology demonstrations, creative stress-relieving toys, and interactive art installations, demonstrating its broad adaptability. Attached Figure Description

[0021] Figure 1 This is a structural schematic diagram of Embodiment 1 of the present invention; Figure 2This is another structural schematic diagram of Embodiment 1 of the present invention; Figure 3 This is a schematic diagram of the structure of Embodiment 2 of the present invention; Figure 4 This is another structural schematic diagram of Embodiment 2 of the present invention; In the diagram, the elastic body is 1 and the force-applying structure is 2. Detailed Implementation

[0022] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments. Example 1

[0023] like Figure 1-2 As shown, this embodiment is a simple and low-cost two-dimensional recursive deformation teaching aid or toy. The elastic body 1 is a ring made of a superelastic NiTi shape memory alloy wire. The stable configuration of this ring in a stress-free state is a standard closed circle, i.e., the first closed form. The alloy wire diameter is approximately 2mm, and the outer diameter of the ring is approximately 12cm. The force-applying structure 2 consists of two symmetrical silicone sleeves covering the outer circumference of the shape memory alloy ring. The silicone sleeves are approximately 3cm long, closely fitting the metal wire to provide good friction and a comfortable grip, while also marking the position of the initial force application point. The working process of this structure is as follows: The user holds the two silicone sleeves with both hands and applies a pair of torques in opposite directions, such as turning counterclockwise with the left hand and clockwise with the right. Under the action of torque, the ring undergoes plastic elastic deformation, gradually concave in the middle, and finally forms a stable planar figure-eight structure, that is, the first-level hourglass shape, with a narrow neck in the center and bulges at both ends.

[0024] After maintaining or relaxing the overall shape, the user can select any enlarged portion of the figure-eight shape and treat it as a new loop. Apply a reverse torque to both sides of this enlarged portion. This causes the enlarged portion to neck and deform into a smaller, secondary figure-eight shape. Repeating this recursive operation generates a chain-like structure composed of interconnected small figure-eight units on the initial loop.

[0025] After ceasing the applied force and allowing the device to stand still or be slightly adjusted, due to the superelasticity of the shape memory alloy, the elastic body 1 will automatically return to its initial ring shape, ready for the next deformation demonstration. This structure is extremely simple, and the deformation process is clear, stable, and highly reversible. It perfectly demonstrates the complete topological transformation process from a two-dimensional closed loop to a figure-eight shape, and then to a recursive chain structure. Example 2

[0026] like Figure 3-4As shown, this embodiment provides a more visually appealing and dynamic three-dimensional recursive deformation device. The elastic body 1 is a sealed spherical capsule made of highly transparent and flexible food-grade silicone through a rotational molding process. The capsule has a diameter of approximately 10 cm and a uniform wall thickness. Its interior is sealed and filled with colored fine sand, accounting for approximately 40% of the total volume, and is either evacuated or filled with a small amount of inert gas. The force-applying structure 2 consists of valves at the top and bottom ends of the capsule, flush with the capsule itself. This ensures that each end of the hourglass has a valve, facilitating the expansion of the smaller hourglass after recursion. Before bidirectionally rotating the capsule, a thin steel needle is inserted through one valve, penetrating the entire capsule and extending from the other valve. When the capsule is bidirectionally rotated, the entire capsule deforms into an hourglass shape around the steel needle. The narrow neck in the middle encloses the steel needle, naturally forming a narrow opening. The steel needle can then be removed from the funnel. The working process is as follows: The user applies a counter-rotating torque to the balloon. The balloon is gradually tightened in the middle, forming a three-dimensional hourglass shape. Under the influence of gravity and pressure difference, the colored sand inside slowly flows from the upper chamber through the narrow neck to the lower chamber, creating a dynamic quicksand effect. Depending on whether the torsion angle increases or decreases, the size of the narrow neck opening also decreases or increases, thereby controlling the falling rate of the colored sand inside.

[0027] Once the hourglass shape stabilizes, the user can attempt to recursively manipulate one of the chambers. Pinch the balloon walls on either side of the center of the chamber with your fingers; slight indentations or textures can be designed here as guides to apply a reverse torque. It can be observed that under the action of localized torque, the shape of the chamber changes, the internal sand flow path is affected, and the overall structure evolves into a more complex multi-chamber shape. Due to the high complexity of the three-dimensional balloon deformation, precise force control is required to form a complete secondary hourglass shape.

[0028] Finally, by releasing all torque and inverting or gently shaking the device, the spherical capsule will eventually return to its initial spherical shape under the combined effects of material elasticity and the flow of internal sand grains. This immersive sensory experience integrates visual, tactile, and morphological changes. It visually demonstrates the coupling relationship between torque, elastic deformation, and fluid dynamics, dynamically illustrating the philosophical concept of generation and flow. Example 3

[0029] This embodiment is a modular device for classroom teaching or precision demonstrations, capable of displaying multi-level recursive structures more clearly and stably. The elastic body 1 is composed of multiple independent sub-units connected in series. Each sub-unit is a rigid ring-shaped skeleton injection-molded from a memory polymer, externally covered with an elastic silicone layer, forming a reinforced elastic ring. The stable configuration of the sub-unit is a circular ring. Adjacent sub-units are connected in series via movable connectors such as micro universal joints or short chains. This connection allows for limited relative rotation and displacement of adjacent rings in multiple directions, but primarily transmits torque. The force-applying structure 2 is directly integrated into each sub-unit. Two raised corrugated gripping areas or differently colored silicone blocks are symmetrically arranged on the outer periphery of the elastic layer of each sub-unit as dedicated force-applying points for that sub-unit.

[0030] Initially, the toroidal surfaces of all sub-units are roughly aligned, giving the overall appearance of a thick closed ring or multi-segmented ring. The user grasps the gripping areas on two symmetrical sub-units located in the center of the whole and applies a reverse torque. Due to the transmission of torque through the connectors, multiple sub-units deform in tandem, and the entire modular system evolves from a thick ring shape into a large figure-eight hourglass shape, with the narrow central section consisting of several tightened sub-units.

[0031] At this point, a group of sub-units constituting a certain bulge of the hourglass becomes an operable modular group. The user can precisely select a specific sub-unit within this modular group and apply an independent, localized reverse torque to it. This operation will cause the sub-unit to deform into a small figure-eight shape on its own scale, thus clearly inserting a secondary hourglass unit into the entire chain structure.

[0032] By selectively performing recursive operations on sub-units at different levels and positions, a multi-level chain system with a clear structure and hierarchy can be gradually constructed and can stably maintain this form, making it easy to explain and observe step by step.

[0033] In summary, the principle of this embodiment is as follows: by applying a pair of opposite torques to a specific point of an elastic body 1 that is topologically closed, the body can be induced to undergo symmetry breaking and necking, thereby evolving from an initial homogeneous closed form into an hourglass-shaped structure with a central narrow neck and two end bulges. Furthermore, by recursively applying this generation logic of applying opposite torques to induce necking differentiation to any newly formed bulge, the same morphological differentiation process can be reproduced at this local scale, generating secondary hourglass-shaped subunits. By repeating this recursive operation, a complex chain structure that presents a double-helix chain pattern on a two-dimensional projection can be constructed, consisting of multi-level hourglass units connected sequentially through their narrow necks. Thus, a controllable recursive generation mechanism is realized in the physical entity, driven by a single, repeatable mechanical operation, from simple to complex and from homogeneous to hierarchical forms.

[0034] The specific embodiments described herein are merely illustrative of the spirit of the invention. Those skilled in the art to which this invention pertains may make various modifications or additions to the described specific embodiments or use similar methods to substitute them, without departing from the spirit of the invention or exceeding the scope defined by the appended claims.

[0035] Although this document frequently uses terms such as "elastic body 1" and "force-applying structure 2," the possibility of using other terms is not excluded. These terms are used merely for the convenience of describing and explaining the essence of the invention; interpreting them as any additional limitation would contradict the spirit of the invention.

Claims

1. A method for recursively generating hourglass-shaped and chain-like deformation structures, characterized in that, Includes the following steps: S1: Provide a flexible body that presents a first closed form in its natural state; S2: Determine two initial force application points on the flexible body; S3: Apply a pair of torques in opposite directions to the two initial force application points to drive the flexible body to deform from the first closed form to the first hourglass form, which has a narrow neck in the middle and two bulges at the ends. S4: Take any one of the enlarged parts in the first-level hourglass shape as a new working area and apply a new round of torque in the opposite direction to it, so that it recursively deforms into a smaller-scale second-level hourglass shape. S5: By repeating the above recursive steps, a chain structure is formed by connecting multiple hourglass shapes sequentially through their narrow necks.

2. The method for recursively generating hourglass and chain-like structures according to claim 1, characterized in that, The first closed shape is a two-dimensional closed loop, and the hourglass shape is a planar figure-eight shape; or the first closed shape is a three-dimensional sphere, and the hourglass shape is a three-dimensional hourglass shape.

3. The method for recursively generating hourglass and chain-like structures according to claim 2, characterized in that, When the first closed form is a three-dimensional sphere, it contains a flowable medium; during the deformation process, the flowable medium flows between the cavity formed by the expanded part and the narrow neck.

4. The method for recursively generating hourglass and chain-like structures according to any one of claims 1-3, characterized in that, The chain structure formed by at least one recursive deformation presents a double-helix chain spatial pattern on a two-dimensional projection.

5. An apparatus for implementing a recursive method for generating hourglass and chain-like structures as described in any one of claims 1-4, characterized in that, include: The elastic body (1) is made of a flexible material and its inherent stable configuration is a closed loop or a spherical shape; The force-applying structure (2), fixed to or formed on the elastic body (1), is used to allow the user to apply torque in the opposite direction to form a narrow neck.

6. The recursive generation device for hourglass and chain-like structures according to claim 5, characterized in that, The elastic body (1) is a ring structure made of shape memory alloy or polymer with shape memory properties.

7. The recursive generation device for hourglass and chain-like structures according to claim 5, characterized in that, The elastic body (1) is a sealed capsule made of transparent flexible material and is filled with a fluid object in the form of particles or liquid.

8. The recursive generation device for hourglass and chain-like structures according to claim 7, characterized in that, The force-applying structure (2) consists of two valves located at opposite poles of the sealing bladder.

9. The recursive generation device for hourglass and chain-like structures according to claim 5, characterized in that, The elastic body (1) is composed of multiple annular or spherical sub-units connected in series by movable connectors, which allow relative rotation between adjacent sub-units.

10. The recursive generation device for hourglass and chain-like structures according to claim 5, characterized in that, The force-applying structure (2) includes two gripping parts symmetrically arranged on the elastic body (1), or a specific action area marked by color, texture or physical indentation on the surface of the elastic body (1).

Images