Threaded fastener, parametric design method and multidimensional applications based on six-dimensional topological projection constant

Abstract

This invention discloses a threaded fastener based on a six-dimensional topological projection constant, a parametric design method, and multi-dimensional applications. Its core lies in applying the six-dimensional spherical S... 6 The natural geometric constant projected onto a two-dimensional cross-section—a 1-radian angle (α = 180° / π rad, abbreviated as R1 angle)—is used as the tooth profile angle, replacing existing three-dimensional empirical angles such as 60° and 55°. In this thread structure, the sum of the half-angles on the bearing side and the non-bearing side is always α, the pitch P(z) varies according to ΔP(n′) = K∫F e^(−km)dm, and the root radius R(z) adaptively adjusts with the pitch. This invention further reveals that the operating logic of modern computers is derived from the pattern plate principle of a jacquard loom, but lacks the strict six-dimensional symmetry of the loom body; by introducing α = 1 rad, this symmetry can be restored in chip wiring, achieving curvature continuity at interconnect corners and solving the problems of high-frequency signal reflection and crosstalk. This topological constant is widely used in bionic implants, exoskeletons, bridge water dividers, and turbine guide vanes to achieve continuous force flow transmission and vibration energy dissipation, providing a unified natural geometric benchmark for cross-scale engineering design.

Description

[0001] This invention relates to the field of mechanical structures and connection structures, specifically to a single-arc tooth profile structure with a specific arcuate contour, and the application of this single-arc tooth profile structure in mechanical transmission, connection, positioning, load bearing, and force transmission structures. It also relates to threaded fasteners and their parametric design methods, as well as engineering structural design principles based on natural geometric constants. Specifically, it relates to a threaded structure with a tooth angle of 1 radian (α = 180° / π rad), and its applications in bio-adaptive scenarios such as medical devices, bionic implants, exoskeletons, and surgical robots; applications in fluid and vibration environments such as bridge water distribution structures, turbomachinery, and vibration-resistant components; and applications in microelectronic devices such as integrated circuit chip interconnect layouts and high-frequency signal transmission structures. Background Technology

[0002] Existing engineering structural design has long been constrained by the empirical paradigm of three-dimensional Euclidean space, mainly in the following aspects: In the field of mechanical connections, known thread profile angles (such as 60°, 55°, 30°, etc.) are all subjective empirical values ​​based on three-dimensional space and have no natural integer relationship with radians. This not only leads to degree / radian conversion errors and programming ambiguities in CNC systems and robot controllers, but also makes it difficult to maintain a consistent mechanical response across different scales due to the lack of a high-dimensional geometric origin. In medical devices and implants, empirical thread profile angles are difficult to match the natural angular characteristics of human tissues and physiological movements, easily causing problems such as stress concentration, interface micro-movements, and insufficient long-term implantation comfort. In vibration-resistant structures such as bridges and turbines, existing thread profile angles are also not designed based on the optimal angle for natural flow distribution and energy dissipation, resulting in limitations in fatigue resistance and hydrodynamic performance.

[0003] In the field of microelectronics and chip design, interconnects, vias, and active areas of devices commonly employ right-angle (90°) or empirically determined bevel angles (such as 45° or 60°) routing schemes. This Cartesian coordinate-based partitioning method is prone to impedance discontinuities, signal reflections, and increased crosstalk due to geometric abrupt changes in high-frequency signal transmission. In nanoscale high-density wiring, the concentrated current density at right angles or empirically determined bevel angles accelerates electromigration failure. While existing EDA tools support routing at arbitrary angles, they lack unified geometric constant guidance, and the design process still relies on empirical iterations.

[0004] Lack of theoretical foundation: Whether it is macroscopic mechanical structures or microscopic electronic devices, existing designs are not based on mathematical foundations that are isomorphic to natural geometric structures. Although empirical parameters have been optimized over a long period of time, they are essentially "trial and error convergence" rather than "mapping of natural laws," making it difficult to achieve a unified design paradigm across scales and fields.

[0005] Therefore, there is an urgent need for a universal design principle that originates from natural geometric constants, has a unified numerical system, and combines biological adaptability, vibration resistance, and high-frequency electrical performance. Summary of the Invention

[0006] The purpose of this invention is to provide a threaded fastener and a parametric design method, and further extend this design principle to the field of chip circuit design. Its core lies in using 1 radian (α = 180° / π rad, abbreviated as R1 angle) as the topological projection constant of the two-dimensional engineering interface. This constant originates from the six-dimensional sphere S. 6 The canonical projection onto a two-dimensional section is not geometrically equivalent to a three-dimensional empirical angle.

[0007] I. Structure of Threaded Fasteners A threaded fastener having an axial cross-sectional tooth angle α = 180° / π rad (1 rad); The half-angle β1 of the bearing side and the half-angle β2 of the non-bearing side satisfy β1 + β2 = α; The pitch P(z) is designed along the axial direction according to the following variation law: ΔP(n′)=K∫0ᴺF e^(−km) dm Where K is the proportionality coefficient, F is the axial load, k is the attenuation coefficient, and m is the integral variable; The radius of the root arc R(z) is adaptively adjusted according to the pitch: R(z)=kᵣ·P(z), where kᵣ is the arc coefficient.

[0008] II. Parametric Design Method This invention further provides a parametric design method for the above-mentioned threaded fasteners, including: based on a six-dimensional spherical S 6 The SO(7) symmetry group constraint is used to determine the tooth angle α = 1 rad; the distribution ratio of β1 and β2 is determined according to the mechanical requirements and biocompatibility requirements of the assembly interface; and the axial distribution law of P(z) and R(z) is optimized by combining the load spectrum and material properties.

[0009] III. Multi-dimensional application scenarios and beneficial effects 1. Medical devices and bionic implants like Figure 2 As shown, the R1 thread of this invention is applied to brain-computer interface electrode fixation, dental and orthopedic implants, exoskeleton joint connections, and surgical robot end effectors. The 1 rad tooth angle closely matches the natural force angle of human tissue. Combined with variable pitch and adaptive tooth root arc, it achieves continuous force transmission, significantly reduces stress concentration, and improves implantation comfort and long-term stability. Taking the alligator-like tooth implant as an example, the R1 thread body (2) and the alveolar bone tenon / bone tissue (3) form a biomimetic interlocking structure (see Figure 2Its tooth angle is consistent with the natural growth angle of biological teeth. Combined with the variable pitch design, it can achieve a self-locking effect of "the more you press, the tighter it gets", which significantly improves the pull-out resistance and torsion resistance.

[0010] 2. Vibration-resistant and fluid engineering structures The R1 thread of this invention is also applicable to structures subjected to fluid impact or strong vibration loads, such as bridge water dividers, turbine guide vanes, and high-speed rotating shaft systems. The 1 rad thread angle is consistent with the stable flow divide angle formed by long-term evolution in nature (such as the water-facing sharp angle of traditional boat-shaped bridge piers in the mountainous areas of southern Anhui, with a measured average value close to 1 rad), which is conducive to smooth fluid flow and vibration energy dissipation, and improves the fatigue resistance and vibration resistance of the structure.

[0011] 3. Integrated Circuit and Chip Circuit Design (Core Extension) The α = 1 rad topological constant described in this invention can be extended to the circuit design principles of integrated circuit chips to resolve the inherent contradiction between high-frequency signal transmission and high-density wiring.

[0012] Traditional chip wiring often uses 90° or empirical bevel angles, which easily lead to signal reflection and crosstalk. This invention proposes using α = 1 rad as the intrinsic topological angle constant for the chip's two-dimensional wiring layer, applicable to interconnect corners, T-junctions, cross junctions, via transition regions, and high-frequency differential pair wiring. The principle is that a 1 rad bevel angle is S... 6 The natural projection of the SO(7) symmetry group onto a two-dimensional interface has curvature continuity, which can make the signal current path closer to the natural force flow line and reduce electric field distortion and impedance abrupt change.

[0013] Furthermore, the parameterization concept of variable pitch P(z) and adaptive root radius R(z) described in this invention can be mapped to the design of gradual linewidth transition and corner radius rounding in chip wiring: along the signal transmission direction, the linewidth is adaptively adjusted according to the equivalent current load function, and a transition radius proportional to the local linewidth is introduced at the corner, thereby achieving a unified "force flow-current" conduction law at the nanoscale. This design paradigm is naturally compatible with the radian system of mathematics, facilitating topology optimization using EDA tools, eliminating angle-to-radian conversion errors, and providing a new path to overcome the interconnection bottleneck in the post-Moore's Law era.

[0014] 4. Numerical Control and Robotics Systems The R1 tooth profile angle, defined using radians, is naturally consistent with the core mathematical language of CNC systems and robot controllers, eliminating degree / radian conversion errors, simplifying programming logic, and improving assembly accuracy and control stability.

[0015] In summary, this invention replaces existing empirical design parameters with a 1-radian topological constant, mapping six-dimensional topological symmetry from macroscopic mechanical structures to microscopic electronic structures, thereby achieving a unified design paradigm across scales and domains.

[0016] Explanation of the origin of geometry The tooth angle α = 180° / π rad (1 radian angle) described in this invention is mathematically derived from the algebraic realization of a six-dimensional spherical topological structure, specifically constructed as follows: We introduce a six-dimensional generalized form of the Luo Shu matrix, M6∈Mat(6,R). The elements of this matrix are natural numbers from 1 to 36 arranged according to the symmetry of a six-dimensional magic square, satisfying invariance in the summation of rows, columns, diagonals, and the hyperplane. We then apply M6 to the six-dimensional Euclidean space R. 6 Medium unit sphere S 6 ={x∈R 6 The projection operator P of the orthonormal basis of ||x|| = 1} yields the projection angle invariants: α=arccos(tr(PM6Pᵀ) / ∥M6∥)=180° / π rad This derivation relies solely on the discrete symmetry of the six-dimensional magic square group and S. 6 The continuous symmetry compatibility constitutes a specific algebraic expression of the six-dimensional topological mapping.

[0017] Furthermore, the aforementioned algebraic invariants can be rigorously verified using a six-dimensional differential geometric structure. The six-dimensional unit sphere S... 6 Having the SO(7) rotational symmetry group, the metric induced by its Killing form has a constant cutoff curvature on any two-dimensional section Σ. Calculations using Lie group representation theory show that this angle invariant remains strictly unchanged under the influence of the SO(7) group, with a uniquely determined value of 180° / π rad. The above projection process is completely independent of the Cartesian coordinate system of three-dimensional Euclidean space R³. Existing empirical tooth profile angles such as 60° and 55° are subjective settings within three-dimensional space, and their values ​​do not satisfy the S... 6 The symmetry constraints of a manifold cannot be obtained through any topological derivation in R³, and the two are not geometrically equivalent.

[0018] As a natural confirmation of the projection constant of a six-dimensional spherical topology onto a two-dimensional cross section, two typical examples can be observed: Firstly, the traditional boat-shaped bridge piers (water-dividing peaks) remaining in the streams of the southern Anhui mountains: their water-facing sharp angles are mostly distributed in the range of 45° to 58°, with an average value close to 1 rad (57.3°), rather than the empirical value of 60°.

[0019] Secondly, at the microscopic scale, the optimal signal transmission path in integrated circuit chips also tends towards continuous curvature turning. Combined with... Figure 1 tooth shape superposition comparison and Figure 2 As can be seen from the biomimetic assembly diagram, the α = 1 rad of this invention is not an empirical optimization for a single field, but a unified application of the same topological projection constant on low-dimensional engineering interfaces, and its application covers macroscopic mechanical components and microscopic electronic devices.

[0020] Figure 1 This invention presents a comparison of the tooth profile and parameters of the R1 thread and the standard 60° empirical thread in the axial section. The left figure is based on a six-dimensional spherical S... 6 The R1 thread with topological projection constant (α = 1 rad ≈ 57.2958°); the right figure shows a standard 60° thread subjectively defined based on three-dimensional Euclidean space. The two have different geometric origins and are not geometrically equivalent.

[0021] Figure 2 : The overall structure and assembly diagram of the crocodile-like tooth implant of the present invention. It shows the biomimetic interlocking relationship between the threaded body (2) with a tooth profile angle of α=1 rad and the alveolar bone tenon / bone tissue (3), and the reference numeral (1) is the matching abutment or fastener.

[0022] Figure 3 : A comparative diagram of chip interconnect corner structures. The left image shows a traditional 90° or 45° wiring corner, which has an electric field distortion region; the right image shows a wiring corner based on a topological constant of α = 1 rad, which, combined with a gradual change in linewidth and a circular transition, achieves a uniform electric field distribution.

Claims

1. A threaded fastener based on a six-dimensional topological projection constant, characterized in that, Its axial section tooth profile angle α=180° / π rad; the bearing side tooth profile half angle β1 and the non-bearing side tooth profile half angle β2 satisfy β1+β2=α; the pitch P(z) is distributed along the axial direction according to the following range law: ΔP(n′)=K∫0ᴺF e^(−km) dm, where K is the proportional coefficient, F is the axial load, k is the attenuation coefficient, and m is the integral variable; the root radius R(z) is adaptively adjusted with the pitch: R(z)=kᵣ·P(z), where kᵣ is the radius coefficient.

2. The threaded fastener according to claim 1, characterized in that, The α = 180° / π rad is the six-dimensional Euclidean space R. 6 Medium unit sphere S 6 The topological constants projected onto the two-dimensional section are determined by the six-dimensional lattice matrix M6∈Mat(6,R) acting on S. 6 The standard orthogonal basis projection operator P is obtained, and the calculation formula is: α=arccos(tr(PM6Pᵀ) / ∥M6∥)=180° / πrad.

3. The threaded fastener according to claim 1, characterized in that, The α = 180° / π rad is the six-dimensional sphere S. 6 The curvature invariant of the SO(7) symmetry group on a two-dimensional section is not geometrically equivalent to the empirical tooth angle in three-dimensional Euclidean space.

4. The threaded fastener according to claim 1, characterized in that, It is used in medical devices, bionic implants, or exoskeleton connection structures.

5. The threaded fastener according to claim 1, characterized in that, Vibration-resistant components used in bridge water distribution structures, turbine guide vanes, or high-speed rotating shaft systems.

6. An engineering interface design method based on six-dimensional topological projection constants, characterized in that, The intrinsic topological rotation constant α of the two-dimensional engineering interface is determined to be 180° / π rad; the value of α originates from the six-dimensional sphere S. 6 The standard projection invariant of the SO(7) symmetric group on a two-dimensional section, or derived from the action of the six-dimensional Rosic matrix M6 on S. 6 The algebraic invariants of the standard orthogonal basis projection operator; the two-dimensional engineering interface includes mechanical connection interface, fluid diversion interface or integrated circuit wiring interface.

7. A six-dimensional topology chip wiring method based on the principle of homogeneity of jacquard patterned plates, characterized in that, Includes the following steps: (1) The six-dimensional topological projection constant α = 180° / π rad is used as the intrinsic rotation angle of interconnect corners, T-junctions, cross junctions and via transition areas; (2) Based on the six-dimensional dynamic symmetry of the jacquard loom, the existing wiring topology based on two-dimensional pattern board logic is modified so that the geometric configuration of the signal transmission path matches S. 6 Curvature continuity of manifolds.

8. The method according to claim 7, characterized in that, The six-dimensional dynamic symmetry specifically refers to the fact that the loom system has a strict six degrees of freedom. The operating logic of modern computers originates from the two-dimensional pattern control principle of jacquard looms. This method introduces a topological constant of α = 180° / π rad to restore the six-dimensional symmetry of the loom body in chip physical design.

9. A chip interconnect structure, characterized in that, The geometric angle of at least some interconnect corners, junctions, or transition regions is α = 180° / π rad; the interconnect structure adopts a line width gradient design, and adaptive circular arc transitions are provided at the corners.

10. Application of a tooth angle of α = 180° / π rad in the preparation of stress-concentration-resistant and highly biocompatible medical devices.

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