Dimension-shifting machinery

Abstract

Dimension-shifting machines are holographic models of polytopal devices. They are unique for having embodied images and dynamic holograms rolled into ones, thereby facilitating physical dimension reduction. To soundly 'flatten' a polytopal device, ridges become the dominant building blocks (11). Operative constructions of building blocks are called motifs, whereas pending ones are imperfect motifs (12). They obey two simple rules: Ⅰ. Take turns to take shape with time; Ⅱ. Take shape to take turns in space. These rules incorporate reconstruction intervals and operation intervals, respectively, into the lowered space. A temporal arrangement of such intervals will manifest itself as a dynamic arrangement of gaps (9-9) around the motifs concerned. Overlaying this spatiotemporal manifold with an arrangement of spectral attributes (15 / 16, 16 / 17, 15 / 17) can preserve image-hologram reversibility besides motif-motif relations. A spatiotemporospectral manifold unified like this constitutes the required machinery to model in real time while shifting dimensions on the fly.

Description

Dimension-Shifting Machinery 易維之機

[0001] Spacetime science often appears to be too formidable and intangible to understand as the subject is literally mind-bending within a curved spacetime. The holographic principle [NPL1] helps by providing a 'flat' for a bent mind to think straight.

[0002] Feasibility has been shown not only on paper but also with a toy known as Melinda's 2x2x2x2 and a few derivatives. These tours de force emulate the states of some higher-dimensional twisty puzzles physically in the everyday world.

[0003] To shift dimensions bona fide, however, path emulators such as the present invention are indispensable.

[0004] Twisty puzzle toys are excellent devices -- objects or machines -- to start modelling on because their sequential moves premise an existing temporal dimension and allow insertions of other intervals between moves as needed.

[0005] Moreover, the inevitable manoeuvres of breaking-mending in / on the flat match the inherent abilities of their dissected bodies. Once 'flattened', these bodies, whether sitting still or permuting, are ready to run like machines.

[0006] Furthermore, relational information must be preserved across different dimensions by having the building blocks somehow coded, which is the built-in feature of their systematically coloured pieces.

[0007] Ultimately, theory is something, but practice is everything. To enhance the world understanding of something profound, what can be better than some widely available toys that offer hands-on experience?

[0008] The current treatment takes no account of any difference between in / on the flat. Although zero effective gravity is assumed in all descriptions, nonzero influence in practice can be counteracted without departing from the spirit of the art.

[0009] The present disclosure relates to non-static N-dimensional (ND or N- for short) models for physics, geometry, and games, where N is any natural number greater than 2 in practice. The development is based on holographic principle and involves spatiotemporal or spatial and temporal manipulation technology with the aid of variable spectra in the sense of divided lights or codes. Embodiment in the nature of twisty puzzles will be described.

[0010] In 2017, Melinda Green, the proprietor of Superliminal Software and a developer of MC4D for 4D twisty puzzle simulation, disclosed her invention called Melinda's 2x2x2x2 [NPL2]. This was considered the first proof of concept that operating a 4D device physically in 3-space was possible. It emulated the states of permutable 3D stickers found on a 2x2x2x2 simulated hypercube, a 4D analogue of 2x2x2 Pocket cube [PTL1]. By abstracting the 4c or four colour-coded 3D stickers from each 4D cubie or constructional piece, the dissected hypercube was transformed into sixteen tetrahedra with colour-coded apices. Based on this, Oskar van Deventer, a prolific puzzle designer, rendered a radially symmetric 3D model for her in 2014, using Reuleaux or pillowed tetrahedra [NPL2]. She came up with a new idea, though. Since alternated cubes were tetrahedra, an inverse process called augmentation or apiculation would convert tetrahedra back to cubes with isosceles-right-tetrahedral or orthogonal-triangular-pyramidal apices as superficial quarter cubes. By means of topology, Melinda delivered to the world a 4D puzzle in an ingenious 2×(2x2x2) cuboid.

[0011] In 2022, Grant Staten, Luna and Hactar extended it to 3x3x3x3 [NPL3] after progressively adding 3c, 2c and 1c cubies [NPL4] and invented their 4D analogue of 3x3x3 Rubik's cube [PTL2]. A physical 3x3x3x3 simulator, which would handle any lengthy "canonical move" in one virtual step, was soon released by Akkei [NPL5].

[0012] These physical puzzles use canonical movesets to ensure emulated states correspond to MC4D-simulated ones. Their free turns in unison resemble those of reusable-connector puzzles such as Triamid [PTL3].

[0013] The disclosure contains four subsections: a summary of the invention, technical problems, solutions to the problems, and advantageous effects of the invention.Summary

[0014] Dimension-shifting machines are holographic models that mainly emulate under physical dimension reduction the path evolution of N-polytopal devices.

[0015] To preserve the N-spatial information, (1,N−1)-spatiotemporospectral manifolds are devised. They are unities of one temporal component and N−1 spatial components together with some device-dependent spectral attributes.

[0016] Temporal and spectral arrangements manifest themselves as dynamic spatial arrangements, i.e. the spatial arrangement is complemented and overlaid, respectively.

[0017] Manipulations of the manifold rely on some key and recurrent structures in spacetime called motifs, whose operative constructions reflect device operation modes, if any.

[0018] From the motifs' (1,N−1)D perspective, the dominant building blocks are (spectral) ridges or (N−2)-faces rather than (ciphered) facets or (N−1)-faces of the devices.

[0019] When harnessing the unity, this natural change in native perspective is crucial, more so than the change in dominance, because the degrees of freedom thus gained foster reciprocation. The machinery is driven by the reciprocation between the motifs and their imperfections.

[0020] Imperfect motifs with pending reconstructions and complementary gaps will take turns to take shape with time, incorporating a time sequence of reconstruction intervals into (N−1)-space. Whenever and wherever necessary, spectrally distinct motifs will take shape to take turns in space on purpose and interleave the sequence with operation intervals.

[0021] Spectrally empowered reversibility between (1,N)D embodied image / clone and (1,N−1)D dynamic hologram makes the machinery even more unique.

[0022] The aforementioned constitute the machinery of the machinery. Their makings and workings are alike. They are self-contained and self-consistent since both system and environment are taken into account to compensate for the lost dimension.

[0023] The existing emulator, viz. Melinda's 2x2x2x2, is too tailor-made to do the following.

[0024] a.Emulate paths or correspond 3D turns to 4D twists directly. E.g. gyro movesets that swap the axis of operation contain impossible intermediary states.

[0025] b.Generalize to other dimensions analogously. E.g. an octomino or layout of 2×(2x2) squares is unsuitable for a flattened 2x2x2 cube [NPL3].

[0026] c.Extend to higher orders consistently. New topological categories come with new topological connections, rendering connection schemes inconsistent.

[0027] The knack lies in applying to a manifold repeatedly the spatiotemporal manipulation technology according to the rules set forth above:Ⅰ. Take turns to take shape;Ⅱ. Take shape to take turns.What it does is to extend the manifold from a locally Euclidean-like space under scrutiny to a locally holographic-like spacetime under timely scrutiny, where the word "like" implies a homeomorphism or, in a crude sense, reversibly continuous map.

[0028] Such a topological spacetime is more robust than a topological space. As far as the arrangements of (imperfect) motifs are reasonable from the spacetime point of view, neither breaking-mending, i.e. a switching between spatial and temporal connections, nor gap, i.e. a manifestation of temporal arrangement, will compromise the connectedness of the manifold.

[0029] The spectral attributes preserve the relations between the motifs and the manifold's double role -- a dynamic hologram down in (1,N−1)-spacetime and an embodied image back in (1,N)-spacetime.

[0030] All in all, together with the change in native perspective and the higher degrees of freedom, the dimension-shifting machinery is made flexible, adaptable, and simple enough to solve all the technical problems listed above.

[0031] a.A dynamic hologram is a path emulator, of course, including state emulation. Unlike impossible device states, invalid paths can never evolve unnoticed. So, for the path emulator, all proper 3D turns should correspond to 4D twists directly without venturing outside the state space that may happen to canonical movesets. (Rigid-body) rotations, in particular, of the corresponding device can be emulated through sequences of 3D turns rather than some gyro movesets. Another way to swap the axis of operation is to directly emulate the rotations -- in one spatiotemporally connected piece. This is impossible without a flexible hologram. In fact, the capability of direct emulation is related to something more fundamental than the need for permutation. In a lowered dimension, an imbedded system (a term used explicitly here in contrast to an embedded system in a higher dimension) is inextricable from its environment, which may encompass any relative frame of reference. Since relative motions may be induced by the environment, all path emulators of the devices, including those of still objects, have to become flexible bodies to the extent of animation-compatible. This is one of the unique characteristics of dynamic holograms versus state emulators. In everyday holography, the flexibility of information retrieval under relative motions gives a hologram its hallmark, viz. motion parallax.

[0032] b.A dimensional analogue often hinges upon symmetry rather than geometry. The ridge-and-gap-based holograms are adaptable to any symmetry in the native (1,N−1)-spacetime. Such adaptability ensures a generalization can be materialized analogously. Take a dynamic hologram of the 2x2x2 cube. The three-fold symmetry of each of the 2×2×2 cubies, can be fully-faithfully embodied by an edge-based regular trilateral instead of a face-based convex triambic hexagon with alternating angles [NPL3] or any incompatible shape like monomino / square. The resultant hologram is an octiamond or connected layout of eight trilaterals. For a preview, a slightly exploded view of it is shown inand its perspective 4D analogue in.

[0033] c.An extended hologram is considered a meronomy or hierarchy of sub-parts of the building blocks. In this view, a consistent extension to higher orders can be supported by subsequent flats that may be thought of as the same building blocks but spatially and / or spectrally degenerated. For example, it is a simple task to duplicate the meeting edges, including any spectral attributes at their ends, of each pair of the trilaterals in a polyiamond, while maintaining the original connectivity among them. This is because any trilateral can degenerate quantitatively to edge(s) and then to vertex(es). In contrast to the simplicity offered by the meronomic approach, a taxonomic or discrete-categorical approach introduces various new topological building blocks equipped with diverse topological connection schemes, but possibly with uniform building block shapes if so desired. This latter approach has its own merits but the diversity makes the extension to higher orders analytically challenging and inconsistent, particularly in the connections between discrete categories, such as the 1c-4c cubies of Grant et al.

[0034] Apart from being flexible, adaptable, and simple, the present invention is versatile.i. Multi-function

[0035] Some models may accommodate multiple operation modes and / or levels of difficulty on a single machine when they are configured properly.

[0036] For instance, two apparently distinct 3D puzzles, the vertex-twisting BrainTwist tetrahedron [PTL4] and the face-twisting Pocket cube [PTL1], may possess the same (1,2)D machine hardware. This is possible because the spatially stringent hardware in the lowered space cannot but capture the spirit of the entire equivalence class [NPL6] to which the BrainTwist belongs. Optionally, this very hardware may be further possessed by a third 3D puzzle, a hybrid-twisting SuperX [PTL5, NPL7], so as to boost the difficulty and complexity exponentially. By analogy, a generalization to (1,N−1)D is possible.

[0037] After some minor modifications on the building blocks, e.g. separating each edge into two halves in the (1,2)D case, the machines will also be suitable for an edge-twisting mode or other intermediates between facet-twisting and vertex-twisting modes.

[0038] Nonetheless, it is hard to achieve this level of versatility with a state emulator. Even with a path emulator, it would be impossible to unleash all such potentials if an ND perspective along with more limited degrees of freedom were imposed upon the (1,N−1)D machines.ii. Multi-representation

[0039] In geometry, there are multiple disjointed representations called nets of a polytope. They flatten the unfolded N-polytope to an (N−1)-polytope that is adjoined at selected ridges, lay out the disjointed system of facets in a hyperplane, and dismantle the polytope globally, i.e. all facets at once, by replicating the sub-features or sub-faces of the facets.

[0040] In the current context, there are also multiple locally holographic representations called dynamic holograms of a polytopal device. They imbed the dissected N-polytopal device in a (1,N−1)D machine that is recurrently conjoined at selected cuts, lay out the locally holographic system of motifs on a time-incorporated hyperplane, and reconstruct the device (or its embodied holographic image) locally, i.e. motif(s) after motif(s), by recycling the spectral merons or sub-parts of the motifs.iii. Multi-realization

[0041] In general, a blooming or torsion-free unfolding from a polytope to one of its nets is irreversible due to ambiguity. For instance, a regular octahedral net may yield either a tritetrahedral boat or an octahedral gem [NPL8]. In the presence of spectral attributes, on the other hand, the reblooming of a dimension-shifting machine is unambiguous and thus unlimited. In other words, the reversibility between image and hologram are preserved, so these (1,N)D and (1,N−1)D realizations are equivalent.

[0042] These equivalent realizations, or perhaps realities to some [NPL9], turn out to be in line with the holographic principle. Simply put, it states that the entire information content in a (1,N)D region can be mapped to and from its gravity-free (1,N−1)D boundary.

[0043] Still, there is more to the present invention than meets the eye. With a proper setup designed to translate (N−1)D motions into ND motions, dimension-shifting machines may be imagined to run the devices they emulate, just as quantum simulators do the physics they simulate. Put another way, dimension-shifting machines clone the devices that prescribe the machines, just as genomes encode the proteins that read the genomes. In short, the self-contained, self-consistent, and self-sustaining dimension-shifting machine and the corresponding device may become two sides of the same coin.

[0044] It will be apparent that sufficient descriptions have been given in this application and other modification or embodiment could be made without departing from the scope or spirit of the inventive concepts.

[0045] A 4-dimension-shifting hologram of non-regular tesseract with cuboidal motifs.

[0046] A 3-dimension-shifting hologram of non-regular dual Dino octahedron.

[0047] A 3-dimension-shifting hologram of Okki or dual 2x2x2 and SuperX.

[0048] A 4-dimension-shifting hologram of dual 2x2x2x2 and 4-SuperX.

[0049] In geometry, Schlegel diagrams are distorted representations of an unfilled N-polytope on a hyperplane or (N−1)-space. The word "distorted" implies a loss of consistency, i.e. standard angular and linear measurements of the facets.

[0050] Nets are disjointed representations of the polytope in another hyperplane. The word "disjointed" implies a loss of coherency, i.e. explicit connections between the facets.

[0051] From the ambient N-space perspective, or in a broader sense, from the perspective of a space at least one spatial dimension higher than their projections, they are very useful as all facets and / or their sub-features can be visualized simultaneously.

[0052] From the hyperplane perspective, however, each part of the polytope has to be deduced separately from the ridges. Therefore, both quantitative and relational information becomes imperative. That motivates a full recovery of the losses even at the expense of the facets, which are no longer discernible anyway.

[0053] Spatiotemporospectral manifolds are locally holographic representations of the polytope or similar devices on a time-incorporated hyperplane or (1,N−1)-spacetime. On the one hand, they are meant to accomplish the recovery mission. On the other hand, they are made to emulate the corresponding devices by thinking the loss in space as a gain in time and spectra. In this way, the inconvenience and the disadvantage of the lowered space, e.g. the inevitable manoeuvres of breaking-mending and all those surrounding gaps, can be put to good use.Example 1

[0054] is a (1,3)-manifold of a tesseract or 4-orthotope if non-regular. The building blocks are the ridges (11) of the tesseract, i.e. rectangles. All ridges are isometric to those of the original tesseract. The total of twenty-four ridges or faces (a pair of cubes with 6 faces each plus 12 more faces formed pairwise between the cube edges) of the tesseract remains unaltered.

[0055] The ridges are evenly divided into six imperfect motifs (12). As an example of multi-representation, the 'sleeves' of four ridges incould have been replaced by 'jackets' of four. Some representations may be chiral.

[0056] Whenever and wherever the manifold is scrutinized locally, two adjacent ridges will oscillate timely (9-9) towards the motif concerned (12) and holographically reconstruct the cuboid. In this embodiment, each motif is simply one of the eight ridge-bounded facets of the tesseract. With a view to showing the details, this oblique parallel projection of the manifold is slightly exploded; otherwise, there would have been a cuboidal motif at the 'innermost' facet or pole in the 4D sense without imminent oscillation. The remaining facet or the other pole is composed of the six 'outermost' ridges, which will also oscillate towards each other timely under scrutiny. Likewise, in the 'jacket' representation, the jackets have to be oriented in such a way that all ridges are available for oscillations around a pole.

[0057] All eight motifs will take turns to take shape (with time). All ridges will have their turns to oscillate eventually under rotations with respect to some relative reference frame.

[0058] Each ridge bears a spectral attribute (obverse / reverse: 15 / 16, 16 / 17, 15 / 17), similar to the shiny / dull sides of an aluminium foil, which is one inseparable entity. These attributes divide the remnants of the ciphered, i.e. coded and nulled, facets. Hence, the theoretical total number of combinations is twenty-eight, resulted from choosing two codes out of a spectrum of eight without repeat in any order (8×7×½). But, four pairs of opposite facets of the tesseract never meet, so the total number of distinct spectral attributes is reduced to twenty-four, which is exactly the number of ridges. The surfaces 'inside' the facets are chosen to have like codes in this representation, which could have been flipped to the surfaces 'outside' as a variant. The latter variant gives rise to spectrally conjoined as well as spatially conjoined motifs.

[0059] To perform a rotation on the manifold is easier done than said. First of all, any 4D rotation can be decomposed into principal rotations about six planes (pairwise combination of four axes, 4×3×½), analogous to 3D principal rotations about three axes or 2D rotation about a centre. Take the vertical column and focus on the top sleeve (12) for now. Unfold the sleeve but keep it in one piece so that the topmost ridge with a marked obverse code of (15) can be bumped out by the next lower ridge with an unmarked obverse code of (16). Fold the three remaining ridges over to the adjacent sleeve(s). It is alright to have dangling ridges. Then, roll down all four middle layer sleeves but leave two ridges behind so that the bumping ridge with unmarked (16) can form a new top sleeve that looks like the original one at (12). To form a new bottom sleeve, compare with the original look of the middle layer and unfold any excess ridge(s), and finally wrap around all the dangling ridge(s). The ex-topmost ridge should become the bottommost ridge by now without flipping. The key is to keep everything in one connected piece at all times. After the rotation, the overall appearance should remain intact while the spectral ridges are updated. Four such rotations in a roll yield an identity. The other two out-of-hyperplane rotations of the horizontal rows can be carried out in a similar fashion. The remaining principal rotations are in-hyperplane rotations of the whole manifold about the column and the rows.

[0060] As a whole, this time-incorporated diagram becomes parallel-projected with better mobility and this spectrum-equipped net becomes locally holographic with better reversibility. Now that this dynamic hologram is consistent and coherent enough to shift back to 4D, if a compatible, i.e. greater than 3D, ambient space is available. Two realizations are obvious. An embodied holographic image of the tesseract will be devoid of the cosmetic part of the facets. A clone of the tesseract will have the facets refilled according to the remnants on the spectral ridges and there is no lack of room for creativity here.

[0061] More fun and functions are in store. If an operation time interval is inserted, during which the motif is firstly disengaged, secondly 3D-turned (more freely for a tesseract than a non-regular 4-orthotope), and lastly re-engaged, then the involved spectral ridges will be permuted. Additional operation intervals must be interleaved with reconstruction intervals so that all motifs will take shape to take turns (in space). The entire manifold will be scrambled gradually via cumulative permutations. Some non-random sequences of permutations may be designed to perform intended 'scrambles', such as the principal rotations described above.

[0062] A logical arrangement of reconstruction and operation intervals along with logical arrangements of 3D turns will make the manifold a solvable puzzle. This logical (1,3)-spacetime or 3D turning puzzle will emulate the path evolution of a logical (1,4)-spacetime or 4D twisty puzzle. The current case is parallel to a facet-twisting Tesseract or, in the case of an 'outside' variant, Super Tesseract. A potential 3D analogue is a 1x1x1 [NPL10] or a (Stefan) Pochmann-style Super 1x1x1 [NPL11], which may be retracted from their respective 3x3x3 counterparts with all but the edge cubies blanked out or ignored.

[0063] The natural change in native perspective provides the motifs with additional degrees of freedom for oscillations and enhances the dyadic nature of the ridges with the spectrum. The outcome is a flexible, adaptable, simple, and versatile manifold. The next embodiment is a quintessence of dimension-shifting machinery fortified with such a manifold plus a new twist.Example 2

[0064] is a (1,2)-manifold of a vertex-twisting dual of a cube, i.e. octahedron or 3-fusil if non-regular. Putting the spectral attributes aside, the oscillations (9-9) of the twelve ridges or edges (21), the operations on the six rectangular motifs or edge-bounded faces (22), and the two out-of-plane plus one in-plane principal rotations of the manifold are all dimensional analogues of those in.

[0065] Four U-shaped imperfect motifs, together with two 'inner / outer' motifs, are conjoined by spectral ridges.shows a state when like codes converge. The ridges are dual to the edge of the octahedron. They are organized in such a way that four codes, e.g. {=, ,*,+} within (22), out of a spectrum of eight {=, ,*,+,−,~,#, :} are divided by the quadrants of a rectangular motif, which serves as a vertex of the octahedron in the light of duality. Consequently, 2D-turning one motif is actually 3D-twisting in unison four of the twelve edges about one of the six vertices of the octahedron.

[0066] The puzzle it emulates is the Dino octahedron [NPL12], with some out-of-print dinosaur stickers (only meant for the original cube in the series) as its eponym. In 3-space, this vertex-twisting puzzle may share the same mechanism with the aforementioned face-twisting (Super) 3x3x3. The engineering requirements on the building blocks are substantially different, though. On the contrary, the dynamic holograms of the puzzles in (1,2)-spacetime share the same unity of spatial and temporal components whereas the differences in spectrum and spectral configuration are relatively minor, i.e. eight codes for octahedron versus six codes for hexahedron and transversal versus longitudinal spectral attributes, respectively. In short, dimension-shifting machinery is undemanding all-in-one machinery. What makes the simplification possible is the underlying spatiotemporospectral manifold, which is not some convoluted topological hypersurface, but an orchestrated folding and unfolding of space, time, and spectra.

[0067] The contention can be generalized analogously to other dimensions. For example, each rectangle incan divide four codes by its quadrants with reverse side mirroring, thus contrasting, obverse side. These spectral ridges are dual to the edge pieces of a Dino hexadecachoron or 4-fusil if non-regular. They are organized in such a way that eight codes out of a spectrum of sixteen are divided by the octants of a cuboidal motif, which serves as a vertex of the hexadecachoron in the light of duality. Consequently, 3D-turning one motif is actually 4D-twisting in unison six of the twenty-four edges about one of the eight vertices of the hexadecachoron. The hardware or unity of the resultant manifold is identical to.Example 3

[0068] is a more advanced embodiment as the motifs are conjoined not by ridges but by facets, which are considered ridge-dependent constructions that act in unison or stick together for easier handling as needed. It may seem like a net of the dual figure of the 2x2x2 [PTL1] known as the Okki [NPL13], most likely styled after the octahedral keychain. Both puzzles are isomorphic and topologically equivalent under duality. Duality, as encountered inas well, is frequently exploited in the design of dimension-shifting machines.

[0069] The state of the workhorse rather than the art is the determining factor in deciding a particular 'net'. The dynamic hologram is meant to be a motif-based machine that acquires the '_V_' shape from its horseshoe-shaped motifs rather than any arbitrary net-like configurations of trilaterals, let alone any aesthetic-based showpiece that acquires some 'X' shape(s) merely from symmetry. Certainly, the mirror image ofis an equivalent arrangement, which can be toggled up and down by breaking-mending two trilaterals on both left and right sides. Another useful net-like representation is either one of the two antisymmetric arrangements, which can be obtained by breaking-mending just one trilateral on either side. Nevertheless, a workhorse does not have to be net-like, so there are other meaningful representations, too.

[0070] To directly emulate a rotation, notice how the axes of the machine align with the medians of the dashed triangle (30). A rotation from one axis of operation to the other two amounts to a realignment of respective medians, i.e. by a 120°-turn clockwise or anti-clockwise. It would be quite tempting to do a quick 60°-turn instead, followed by a breaking-mending of a pair of trilaterals to maintain the '_V_' shape. That would be an absolutely valid path evolution involving legitimate states. Alas and alack, the dashed triangle would be either destroyed or reversed, and so would the updated axes. One of the more rigorous and fail-safe ways, especially when the machine is extended to higher orders, is a toggling as in mirroring before a 120°-turn anti-clockwise followed by a breaking-mending of 'the excluded' pair of trilaterals that is lying outside the dashed triangle. To access the other axis, either repeat the above or restart reversely: a breaking-mending of the excluded pair in the opposite direction before a 120°-turn clockwise followed by a toggling.

[0071] There are four placements (31, 32, 33, 34) of conjoined motifs, each comprising four trilaterals. The machinery and all oscillations (9-9) are driven by the reciprocation between the four motifs and two other imperfect motifs. Dialing the left (32) and the right (33) motifs against each other while maintaining the '_V_' shape will deconstruct a motif in exchange for another periodically.

[0072] Axes of operation may be swapped with sequences of turns in lieu of direct emulation. For instance, with primed turns being anti-clockwise,(31′)-(32′)-(33)-(34)-(32′)-(33)-(32′)-(33);(32)-(33′)-(31′)-(32′)-(33)-(34)-(32′)-(33).They correspond intuitively to a half-cube twist, a reconstructive rotation, another half-cube twist, and some final rotation(s), or in other orders. There are various versions and some may be shortened but less intuitive. Routine maintenance via breaking-mending is understood.

[0073] All these complicated manoeuvres and sequences are just for verifying the compatibility between the machine and its corresponding device and demonstrating the rationale behind the operations. In practice, however, the essentials of the puzzle are as simple as (31), (32), (33). No rotation whatsoever is needed. One of the major lessons from lowered spacetime is about move entropy besides move economy. All extravagance with symmetry in N-space becomes a waste of energy in (1,N−1)-spacetime. Inconvenient moves are often self-evidently optional.

[0074] Remarkably, this machine can be overloaded with functions. It may be used as a face-twisting 2x2x2 [PTL1], a vertex-twisting Dino cube [NPL6], and a hybrid-twisting SuperX [NPL7]. The description that follows will focus on SuperX, which basically covers the rest. When a hidden motif composed of a double-layered trilateral facet (36) is turned occasionally, just as encircled by the thin dashed line (and the opposite side alike), the total number of possible positions will increase significantly, from roughly 106to 1022. Perhaps it will be easier if the turns are limited to (36) and (32) plus (33) or if the spectrum is limited to a sub-spectrum.

[0075] Since the conception of SuperX in 1982 [PTL5], it took almost three decades to attain a stable puzzle circa 2011 [NPL7]. On the contrary, they are available instantly in (1,2)-spacetime. This is to emphasize that dimension-shifting machines can become undemanding all-in-one alternatives for demanding counterparts in the original dimensions. It is exactly one of the incentives to try physical dimension reduction.Example 4

[0076] is a 4D analogue of the path emulator depicted in. This chosen representation can be easily compared and contrasted with the state emulator of Melinda's 2x2x2x2 [NPL2] as well as.

[0077] Regarding the former, recall that each evenly coded apex here is morphed topologically to a special tetrahedron or pyramid for a quarter cube. The six tetrahedron edges here fall into the six c4 faces and become invisible diagonal lines perpendicular to the meeting edges of the quarter cubes. Hence, edge-edge connections here become c4-c4 connections while face-face connections here become triple-stickered vertex-vertex connections. Note that these changes in topological connection only apply to c4 cubies, though, as other cubies have other changes.

[0078] Regarding the latter, think ofas a (bent) equatorial plane of. Consistency can be judged from their common spectra. The poles where the new grid patterns of light converge (some with gaps) are on the additional axis. The dashed triangle inis actually a regularly augmented triangle with medians redefined between augmenting vertices and core vertices. It can be generalized analogously to a regularly augmented tetrahedron forwith medians aligned with the four axes.

[0079] The emulator can be generalized to ND for natural N>2 as follows. Convert a 2N-dissected N-cube into its dual, an N-orthoplex, whose net can be laid out flat in a hyperplane of the N-space. The dissection yields 2Nfacets, each being considered a construction of N ridges in the shape of (N−2)-simplex. So, there is a total of 2N×N ridges, doubling the 2N−1×N ridges of the N-orthoplex. Each spectral ridge has N−1 vertices and divides at most N−1 codes out of a maximum spectrum of 2N, i.e. the number of vertices of the N-orthoplex.

[0080] The chosen net(s) should be the one(s) that arranges 'horseshoes' (asymptotically squares for large N) of 2x2=4 composite facets or (N−1)-simplices, such as composite tetrahedra (41, 42, 43, 44) for N=4, on an (N−3)-sphere. Preferably, all horseshoes are connected at the same (second or third) position so that their gaps will be in phase. Each motif is composed of half of all facets in the horseshoe shape, i.e. ½×2N×¼=2N−3horseshoes per motif. Further details are explained according to puzzle types.

[0081] For 2Nand N-SuperX puzzles, each facet twist on the hypercubic device involves half of the puzzle, so it is convenient to designate two motifs on the left-right axis of the corresponding emulator. The N−1 pairs of vertices at which the two motifs meet indicate the half-space of the non-left-right axes. Rotations about the left-right axis are emulated by coordinated turns of both motifs against each other. As other motifs of non-left-right axes cycle through, more motifs will be available for turning. Swapping the axis of operation boils down to rotating a preset regularly augmented (N−1)-simplex, analogous to (30) in. Apart from the direct emulation, relatively short sequences of turns may be implemented for the swaps. The overall structure of the emulator is maintained by breaking-mending. No impossible intermediary states are required. In the current formulation, N-SuperX is nothing more than a capacity on demand by introducing double-layered facets as new motifs, analogous to (36) in. But as a caveat, even the modest N of 4 will result in a vast, or more appropriately, hyper-astronomical number of possible states.

[0082] In the case of N=4, it is noteworthy that the original 3D rendering produced by Oskar van Deventer based on Melinda Green's description of her model actually has all the parts available for a proper assembly. Nonetheless, its tantalizing motifs immobilize reciprocation. On top of that, its highly symmetric arrangement breaks the key oscillations (9-9) apart, disables the temporal component, and hampers the spectral cues. Consequently, the system is deprived of both necessary and sufficient conditions for being a path emulator.

[0083] For 3Nand higher-ordered (MC4D-)simulated puzzles, pick one side of the emulator, say the left side just to be consistent with, and insert a ridge or (N−2)-face between each existing pair of them. Reiterate the insertions but in the next lower dimension with degenerated ridges or merons of ridges so that there are new (N−3)-faces. Telescope the insertions until there are new edges or 1-faces. Then, do the same on the right side including all unpaired ridges. Centroid pieces on the facets of the hypercube correspond to the vertices of the preset augmented (N−1)-simplex, which is positioned mainly on the right side of the emulator except for an apex on the left side. The extents of these additional merons depend on the extents of the additional ridges, which do not have to copy exactly the extents of the original ridges and are never meant to be infinitesimal, so the vertices are not necessarily point-like, either. Higher orders are extended in a similar fashion. For even orders, merons may be deposited symmetrically on both sides. That also implies a split augmented (N−1)-simplex together with numerous split vertices.

[0084] The moves are very similar to those in the 2Ncase but more cumbersome in the presence of more pieces to break-mend and to keep track of. Those middle-(N−1)D-slice moves available to simulated puzzles correspond to emulated moves on (N−2)-spheres and are allowed only when all connected merons are on the spheres. Although the overall spatial arrangement seems curvy, everything remains Euclidean, holographic, and continuous in spacetime.

[0085] As a bonus, unlike a boxy but covert design, the jaggy but overt design here helps visualizing the relations between all the building blocks without peeking or exploding [NPL5].Universality

[0086] The main idea of these examples and of dimension-shifting machinery at large is succinctly highlighted in two long-standing proverbs, modified just to usher in a new era of interaction:‣ Motifs Makyth Machine;‣ Facets Don't Make the Machine.

[0087] All these examples share one common phenomenon of paramount importance: the output number of realizable motifs exceeds any instantaneous number of observable (imperfect) motifs that, in turn, exceeds or dynamically differs from the input number of conjoined (imperfect) motifs. Specifically,has 8 realizable motifs, exceeding by at least 1 motif;has 6, exceeding by at least 1;has 6, exceeding by at least 2;has 8, exceeding by at least 3. Scientifically speaking, when things do not sum up as usual, some interference must be taking place [NPL14]; when seeing is unmistakably less than believing, most probably and highly possibly something non-spatial is in action. The empirical evidence here shows how to tell conjoined motifs and adjoined structures apart and how to tell not only those spatially conjoined motifs but all temporospectrally arranged motifs are in action.

[0088] Inasmuch as spatiotemporospectral manifolds can repeatedly fold and unfold or hide and unhide part of the overall 'lattice', dynamic holograms of dimension-shifting machinery resemble time crystals, in addition to quantum simulators and genomes.

[0089] Dimension-shifting machinery, including the inventive concepts, is expected to offer a wide range of applications, from saving physical storage to probing fundamental reality, and from creating new entertainment to advancing basic science. The undemanding level of sophistication makes it suitable for manufacture on any scale. Business circles, industries, and academia will be benefited from it.

[0090] Manifolds similar tocan be constructed for many polytopes. They will comprise a good portion of practical models for studying geometry and related fields. Although most other embodiment above are elaborated in the nature of toys, it does not mean they cannot be used for other purposes [NPL15].

[0091] At present, both industrial and public interests seem to be in the toy production, which is also an important aspect of the development. Take Melinda's 2x2x2x2, whose exceptional level of difficulty makes the nice-looking puzzle quite otherworldly. Still, sales in over 24 countries worldwide have been increasing since its birth in 2017 [NPL2].

[0092] Based on the disclosed concepts here, a large class of puzzles for everyone is in sight. Their types can be constructional, mechanical, electronic, symbolic, logical, virtual, abstract, etc. The multi-modality will benefit different industries in a sustainable manner, especially those related to education and toys.

[0093] When configured properly, the disguised levels of difficulty can be switched all the way from easygoing to overwhelming. Even identical products can possibly target and satisfy both novices and experts from all age groups. So, the potential market will be as broad as the range of cognitive power.Patent Literature

[0094] US 3655201 (Larry D. NICHOLS) 17 April 1972, abstract and claims 1, 3, 6, & 9. Pattern forming puzzle and method with pieces rotatable in groups.

[0095] HU 170062 (RUBIK Ernő) 28 October 1976, column 3, lines 23 to 31& 44 to 45 and claim 1. Térbeli logikai játék (Spatial logic game).

[0096] HU 207233 B (RUBIK Ernő) 28 March 1991, abstract and claim 1. Elemes térbeli logikai játék (Elemental spatial logic game).

[0097] US 2005 / 0098947 A1 (Charles HOBERMAN, Mathew DAVIS) 12 May 2005, abstract. Transforming puzzle.

[0098] US 4474377 (Johnathan J. ASHLEY) 2 October 1984, abstract and claim 1. Eleven-plane cubical puzzle.Non Patent Literature

[0099] BEKENSTEIN, J.D. Information in the holographic universe. Sci. Am. 2003, 289(2):58-65.

[0100] Melinda GREEN. See superliminal.com / cube / 2x2x2x2

[0101] Grant STATEN, Luna and Hactar. See wiki.superliminal.com / wiki / Physical_Puzzle

[0102] 3c, 2c and 1c cubies. See hypercubing.xyz / puzzles / physical /

[0103] Physical 3x3x3x3 simulator. See youtu.be / QlpCB9ngklE

[0104] BrainTwist and equivalence class. See www.jaapsch.net / puzzles / dinocube.htm

[0105] SuperX. See twistypuzzles.com / cgi-bin / puzzle.cgi?pkey=1910

[0106] MALKEVITCH, J. Nets: a tool for representing polyhedra in two dimensions. See www.ams.org / publicoutreach / feature-column / fcarc-nets

[0107] TALBOT, M. The Holographic Universe. New York: HarperCollins Pub., 1991, p.3.

[0108] 1x1x1. Try rubiks-cube-solver.com

[0109] Pochmann-style Super 1x1x1. Try www.randelshofer.ch / rubik / virtual_cubes / rubik / super_cubes / super_pochmann.html

[0110] Dino octahedron. See twistypuzzles.com / cgi-bin / puzzle.cgi?pkey=6736

[0111] Okki. See robspuzzlepage.com / images / keychain-okki1.jpg

[0112] RӦTHELI, T.F. One plus one equals two: more or less. Theoretical Economics Letters, 2022, 12:972-979.

[0113] CREASE, R.P., MARTIN, J.D., PESIC, P. Physicists at play. Phys. Perspect. 2018, 20:1-3.

Claims

An assembly comprising a plurality of parts whose alternate side(s) at some theoretical limit can only be exposed either simultaneously in the presence of duplicate(s) of said assembly or non-simultaneously in situ, considering both situations from the dimension perspective of said assembly and at least one of said parts has distinct said side(s).A system comprising an input plurality of said assembly of claim 1 to be arranged in such a way that any instantaneous plurality of detectable assembly differs from said input plurality and to be manipulated in such a way that an output plurality of realizable assembly exceeds all said input and said instantaneous pluralities.

Images