Essay Abstract

Study of the physical reality can happen in two different ways that meet each other at a certain point and then complement each other.

Author Bio

Hans van Leunen is a retired physicist. Born in 1941 in the Netherlands. MSc Applied Physics at TUE, Eindhoven Career in Hightech industry. Intensified imaging & software generation Initiator of https://en.wikiversity.org/wiki/Hilbert_Book_Model_Project

Download Essay PDF File

Hans,

Your approach is correct. The reason no one touch that is because it is the domain of Metaphysics. Without using the actual "M" word, this is what I did in my essay.

Marcel,

    J.A.J. van Leunen,

    After reading your essay - I think you may be interested in reading mine...

    The Day after the nightmare scenario - Scott S Gordon

    Thanks for the essay. I fully agree with your statement that "Observations and measurements cannot uncover everything. Only the application of deduction can expose the parts of the physical reality that resist observation. The interplay of measurements and deduction can bring about the necessary confidence. The requirement that experiments must verify everything is sound-ready crap. Much of the physical reality is inaccessible to measurement. In that case, deduction remains the only way of approach". We are 3D rational beings. Our senses and instruments detect only 3D matter-bodies and their interactions. We are unable to appreciate existence and actions in lower spatial systems. We can only infer what happens in lower spatial dimensional systems. Such deductions can bring about continuity of development of universe from its most fundamental entity. For this, conceptual approach will be more suitable than mathematics.

    Regards,

    Nainan

    As I read it your essay outlines the (two) standard approaches to reality - analyzing phenomena or building (synthesizing) models, and suggests a bridge between them. A concrete proposal turns it into a tangible asset. Using the insight that some structures are intinsically finite, that is, there are just a few 'number like' types, is also something I appreciate. There is point of personal interest that you might clarify perhaps: what about octonions? Work on their geometry and/or physics has appeared during the last decade.

    Best.

    a.l.

    Dear friend, As soon as you know that nature applies a Hilbert space as its repository for its data, then you know that it restricts its number systems to division rings. The most elaborate division ring is formed by the quaternions. The octonions and the bi-quaternions are not division rings. Quaternions are optimally suited to store dynamic geometric data in a Euclidean format that combines a time stamp and a spatial location. Observers can retrieve these data, but an embedding field acts as a transporter, such that the observers perceive in spacetime format. The hyperbolic Lorentz transform describes the corresponding coordinate conversion.

    Greathings, Hans

    Dear J.A.J. van Leunen,

    You wrote: "Physical reality clearly has a structure."

    My research has concluded that Nature must have devised the only permanent real structure of the Universe obtainable for the real Universe existed for millions of years before man and his finite complex informational systems ever appeared on earth. The real physical Universe consists only of one single unified VISIBLE infinite surface occurring eternally in one single infinite dimension that am always illuminated mostly by finite non-surface light.

    Joe Fisher, ORCID ID 0000-0003-3988-8687. Unaffiliated

    Hi Hans...

    Your initial statement "The name physical reality is used to display the universe with everything that exists and moves therein." is appreciated as a attempt to clarify semantic interpretive errors... but what fundamentally constitutes Physical?... motion??... some theoretical minimum unit of Energy (QE)???

    What fundamental property do we assign to any one of "everything" to verify the one as Physical?

    If Physical is undefined one can not eliminate the potential for Metaphysical... i.e. other than physical.

    I utilize Metaphysical in my essay to differentiate from Physical "in terms of Physically measureable properties... i.e. sound, light, electromagnetic potential, chemical reaction, quantum particle choreographies, etc.... but are all PHENOMENA processes Physical?" ~ sl FQXi Essay

    If you are speaking of "structure" as being a Spatial construct?... I agree that "deeper layers will reveal "an increasingly simpler structure"... and may I add, that structural form follows function?

    To the degree that one is speaking of a Spatial geometry model, a predicted observation does not constrain the observation to a single geometry model, nor does it necessarily verify all aspects of a theory inferred from a specified "bench" geometry model that resolves the observed prediction.

    That is to say, that direct observation of a gravitational radiation wave, does not verify Einstein's rubber sheet (2D plane) geometry model as the definitive resolve of Space/Energy/Time computational geometry models.

    I do agree that the foundation revealed "must force the development of reality in the selected direction"... i.e. initial conditions are the primary selection criteria of emergence, and in that no Physical Energy Choreographies existed at initial differentiation of Space by Energy, a unique Spatial logic/information quantization by a unified volume unit... i.e. Einstein's concept of a unified 3D field geometry... is a "Good guess"... REF: TOPIC: A Universe Made of Stories by Philip Gibbs

    ... as is the requirement for initial Emission Distribution "equal in all directions from a single point." ~ sl FQXi Essay

    "It is not that something is being created from nothing, it is that undifferentiated potential information is being differentiated as minimum units of Space/Energy/Time/Information" ~ sl FQXi Essay

    If we observe that "the growth process provides restrictions" those restriction must be supported by the underlying "bench" geometry structure utilized as the initial emission/propagation logic.

    That is to say that, to the degree that one is speaking of any process in terms of equations extracted from a specified "bench" geometry model, to verify that the specified "bench" geometry model resolves the observed process, the visual kinematic chain for the theoretical derivation, from the observed geometry, back to the specified "bench" geometry model in which the process is nimated... i.e. given dynamics... must be verifiably unbroken.

    Digital CAD/SIM models can facilitate highly complex visual verification... i.e. if the coded simulation does not visually emulate the observed process, then the Mathematics is not verifiable.

    In that mathematical theories of Space/Energy/Time, at least in principle, are derived from an observed and/or intellectually visualized Spatial geometry model, the quest for a "potential candidate for the foundation of physical reality" is now being pursued at a most fundamental Mathematical level... i.e. as comparative investigations of underlying geometries that support initial Emission Distribution "equal in all directions from a single point".

    REF: UQS Origin Singularity Geometry http://www.uqsmatrixmechanix.com/UQST-TVNH.php

    With permissions supported/granted by the Unified Quantization of Spherical Singularity (UQS) geometry... I have for over 20 years utilized a conventional computer to quantize a unified field geometry environment... i.e. as a Unified Quantization of a Singularity (UQS)... and am currently developing/coding a UQS Virtual Quantum Lab/Game that utilizes a UQS CAD Environment to digitally visual animate, Emission expansion of pulsed minimum units of Energy (QE)... i.e. digital sprites... for distribution/propagation analysis.

    REF: UQS Data Bus http://www.uqsmatrixmechanix.com/UQDB.php

    You state that there are "only three number systems", but all three can be derived from geometry coordinate systems, so why convert the concept of Spatial distribution to a number system, when we now have the digital tools to project the concept directly onto the visual "bench" geometry... i.e. the CAD environment provides "The base model (which) acts as a storage space for dynamic geometric data.", and the SIM interface utilizes timestamped events in the CAD environment to tell a "dynamic story".

    Utilization of CAD/SIM analysis has shown that there is not just one "story that explains why the elementary particle constantly deforms its living space and why the particle possesses a quantity of mass."... i.e. UQS Emission CAD/SIM utilizing an 6 axis Equal Quantization Quaternion CAD environment Space/Energy/Time/Info model, tells the same story.

    I have browsed the Hilbert Book Project, and I admire your directions to resolve a plausible Mathematical model, but you do admit, your numeric conversion process has "become quite complicated", and you have stated simplicity as a verification parameter of foundational fundamentality,

    Not only is it complicated, it has not produced a verifiable mechanism for initial emergence of the minimum units of Energy (QE) required for animating the cosmos.

    I herein reiterate statements from my review of TOPIC: A Universe Made of Stories by Philip GibbsPhilip Gibbs:

    "With regard to "guessing correctly the answer to questions like 'what is "fundamental?'", the "stories" are ancient... REF: TOPIC Indra's Net - Holomorphic Fundamentalness by Cristinel Stoica ... and although such "stories" are prolific, multi-epoch, and multi-cultural, requirement for a logic reduction is a common element."~ sl

    "However, cognitive abilities to resolve a Math model of a logic reduction of "fundamental" are apparently emergent...REF: TOPIC: How to Empirically Confirm a Rational Theory of Fundamentals by Jack H. James ... and application of Math semantics to an invalid logic reduction of "fundamental" have often muddied the cognitive waters."~ sl

    'The Spatial Singularity quantization is fundamental to all derived Spatial relationships/logic. "~ sl

    "In that evolution is driven by the "What is fundamental? quest, then technology is also, and development of digital tools has been reciprocated with a cognitive enhancement, in the form of a more precise Mathematical model of Indra's net "cast in all directions"... i.e. resolve of an Origin Spherical Singularity Geometry, which supports infinite minimum unified volume unit shell closure expansion as a valid CAD environment/field quantization... has now been added to the "stories", and a pulsed Emission of minimum quanta of Energy (QE), is being digitally simulated/animated within the virtual environment.

    REF: UQS Origin Singularity Geometry http://www.uqsmatrixmechanix.com/UQST-TVNH.php "~ sl

    "UQS is not a theory, it is a digital CAD/SIM virtual reality constructed on a logic reduction... i.e. an Origin Emission equal in all Spatial directions from a single point."~ sl

    "Better than a guess?""~ sl

    Hans, without revealing a verifiable property of "Physical", you state that "Only the application of deduction can expose the parts of the physical reality that resist observation."

    Is deduction a Physical Information Energy process?

    "Although they are in many cases stimulated by Physical processes, are mental info processes... i.e. conceptualization, realization, dreams, visualization, intuition, etc.... established on Metaphysical scale bit differentiated streams? "~ sl FQXi Essay

    I agree that "The interplay of measurements and deduction can bring about the necessary confidence." in a theory, or in a digital virtual reality model.

    In that "Much of the physical reality is inaccessible to measurement.", and "In that case, deduction remains the only way of approach.", I highly recommend that one verify the "bench" Origin Singularity geometry form follows function, for a pulsed minimum unit of Energy (QE) Emission, equal in all directions from a single Origin Source, before applying the artifice of the equationist to "everything", or anything, else.

    As per my interpretation of World Science Festival: Topic: "Limits of Understanding" https://www.youtube.com/watch?v=DfY-DRsE86s, the instability w/ current Space/Energy/Time/Info Physics, is as a consequence of the current instability of Mathematics.

    My assessment is that the APPARENT current instability of Math is as a consequence of a geometry coordinate system, in which the fundamental Spatial relationships of the geometry do not support application of minimum units of Energy/Space/Time/Info, to Space/Energy/Time/Info analysis.

    Thanks Hans, for sharing your insights and thus making an opportunity for comment... I would read with attention your comments on my essay entry Title: Knowledge Base (KB) Access as Fundamental to Info Processor Intelligence.

    Will return to rate after I read as many essays as I have time.

    S. Lingo

    UQS Author/Logician

    www.uqsmatrixmechanix.com

      The Hilbert Book Model is a purely mathematical model that starts at a foundation which is an orthomodular lattice. This foundation emerges into a read-only repository that stores all dynamic geometric data of its discrete objects in the eigenspaces of dedicated operators that reside in separable Hilbert spaces that share the same underlying vector space. It is possible to sequence the time-stamps of the archived data and this corresponds with a subspace of the underlying vector space that scans the model as a function of a real number progression value. Observers travel with this scanning window, which represents the current static status quo of the model. Observers can only retrieve data that are archived with a historic time stamp. Elementary particles are the lowest form of observers and they figure in observable events. Further, they are elementary modules that together constitute all other modules and some of the modules constitute modular systems. Some of these modular systems are intelligent species. Two lower categories of super-tiny objects exist that constitute all other objects that exist in the model. These objects are shock fronts and they constitute the interaction between the living space and the objects that live in that space. The living space is a continuum. That continuum transfers information between the discrete objects.

      At higher levels, this model will show resemblance to the environment that is perceived by intelligent species. Only a small part of the model is accessible to the perception by observers. That part may be called physics.

      Hans,

      An interesting new view of the HBM. I still agree the 'interleaved pages' quantum logic view as my 'Red/Green sock trick' essay but this essay gave me a new insight into your use of Hilbert Space.

      Of course I also agree your fundamental tenet, which also matches mine. Start from the simplest level and use deduction. I found it can achieve astonishing results.

      I still haven't properly investigated the validity of quaternions in describing the structure I arrive at, with 2 different & orthogonal momenta pairs (each +/-) varying across 3 degrees of freedom (x,y,z, spherical axes). If you can see any rationale do explain.

      My own rationale and finding is effectively coded by Declan Trail, including working from my recent essays etc. offering a quite sound proof of it's relevance. Of course it arose from the SR rationale so unification is implicit from re-emissions at local c. I'd value your comments from a very different perspective.

      Though rather short I intend to score yours highly as it responds well to the scoring criteria.

      One more matter, I've liked Alan Kadins recent work though he seems to have wandered off track with this one. He concludes; "There is no Hilbert Space"! Can you reconcile any of his argument?

      Nicely written. Very best.

      Peter

        Peter, you can find many highlights of the Hilbert Book Model in http://vixra.org/author/j_a_j_van_leunen. One, in particular, treats quaternionic differential calculus. "Mother of all Field Equations"; http://vixra.org/abs/1709.0324. You might also be interested in "Nature's Basic Dark Quanta"; http://vixra.org/abs/1712.0241.

        Last weekend I visited Vigro in Italy together with 40 students and alumni of the TUE. It was quite enjoyable.

        It is great fun to do this research and discuss with scientists like you.

        Greathings, Hans

        Hi Hans...

        Could you provide a link to a graphic geometry representation of the Origin Singularity of a "purely mathematical model that starts at a foundation which is an orthomodular lattice"?

        Thanks Hans, for making details of your model available.

        S. Lingo

        UQS Author/Logician

        www.uqsmatrixmechanix.com

        Sue,

        The Wikiversity Hilbert Book Model Project contains a slide presentation that treats some highlights of the project

        https://en.wikiversity.org/wiki/Hilbert_Book_Model_Project/Slide_show

        Similarly, the e-print archive http://vixra.org/author/j_a_j_van_leunen contains several documents that highlight aspects of the project.

        The orthomodular lattice is treated in https://en.wikiversity.org/wiki/Hilbert_Book_Model_Project/Relational_Structures. "Diversity of floating platforms"; http://vixra.org/abs/1712.0242 treats the base model that emerges from the orthomodular lattice.

        Sincerely yours, Hans

        Hi Hans...

        There are an infinite number of Orthomodular lattice, any of which can be a Mathematical "foundation"... i.e. coordinate system... on which to derive concepts of Spatial geometry, and any geometry structural "foundation"... i.e. coordinate system... has elements that are more "fundamental"... e.q. origin, axis, dimension, base unit of quantization, singularity etc.... origin being the most "fundamental".

        In that the base unit of the lattice quantization will dictate the form of possible Origin Singularities... i.e. the singular point of Origin enclosure geometry... the base unit of the lattice quantization is a more "fundamental" element of the mathematical "foundation"... i.e. coordinate system... than is the Singularity.

        In a "fundamental" discussion, a distinction must be made between Cubic Singularity Lattice resolve and Spherical Singularity Lattice resolve... i.e. a lattice which resolves a Spherical Singularity is necessary to resolve the "fundamental" logic reduction of an equal distribution in all directions from a single point.

        Utilizing the Wikipedia Ortho Lattice M4 Geometry as the base unit Volume of the "foundation" field quantization... i.e. coordinate system... 6 mutually perpendicular M4 base unit volumes can resolve a Spherical Singularity, ONLY if the relationship of the the M4 base volume unit height = the volume unit width... i.e. any vertices has identical distribution geometry.

        In that Energy is the "fundamental", being distributed onto a Spatial lattice by an Origin Source pulsed emission, and Energy can not be defined as a dimensionless point location, a minimum unit of Energy (QE) must have a Spatial containment/address within the "foundation" field quantization.

        To facilitate EQUAL distribution in all addressable Spatial directions from a single point, that Spatial containment/address must be a UNIFIED minimum unit of Space (QI).

        The UQS field quantization utilizes an M4 Ortho Lattice, in which the relationship of the base volume M4 unit height = the volume unit width... REF: UQS Base Unit Derivation... which resolves 24 UNIFIED minimum units of Spatial (QI) for Origin Source pulsed Energy distribution, of 24 UNIFIED minimum units of Energy (QE), equal in all lattice addressable directions from point of Origin, onto the Lattice... i.e. information structure... in a manner similar to the boot process of a conventional computer. REF: Comparative Singularity Geometry http://www.uqsmatrixmechanix.com/UQST-TVNH.php

        Given h=1, the UQS Origin (0,0,0) expands as 6 equal Emission Axis Sectors which yield 8 Distribution nodes as 3 (x,y,z) coordinates: (-.5,+.5,-.5);(+.5,+.5,-.5);(+.5,+.5,+.5);(-.5,+.5,+.5);(-.5,-.5,-.5);(+.5,-.5,-.5);(+.5,-.5,+.5);(-.5,-.5,+.5), and by Pulse 8 the QE emission distribution resolve log can accumulate resolve intelligence, required to mirror close to the 6 Singularity Shell Emission Nodes (-1,0,0);(0,0,-1);(+1,0,0);(0,0,+1);(0,-1,0);(0,+1,0).

        Subsequently, the UQS coordinate system supports infinite radius expansion of Spherical Shell Closure, in increments of 1 UQS Base Unit Volume height.

        The UQS minimum Spatial unit is a 2D Plane of UNIFIED geometry, defined as 3 (x,y,z) coordinates... e.g. given h=1 the UQS Singularity -Xsector/+Yanchor is a 2D Plane bounded by (-.5,+.5,-.5); (0,0,0); (-.5,+.5,+.5). REF: UQS Information System

        Hans, if you are working with an Orthomodular Lattice and have resolved a Spherical Singularity, it is highly probable we are working with the same "foundation" field quantization... i.e. coordinate system... and in that I am not a physicists, I would like to verify compatibility of your Physics unification concepts with the UQS geometry model.

        I think you will agree that utilization of the Wikipedia image of the M4 Ortho Lattice unit, greatly enhanced my ability to communicate critical elements in this discussion, but if you do not have a visual graphic of your Singularity geometry, I can convert numeric (x,y,z) coordinates to a 3D CAD image, if you provide the Singularity numeric coordinates in relationship to the Origin (0,0,0).

        The UQS CAD Spatial quantization environment satisfies model simplicity criteria... i.e. 4 screens of code can node data populate the "foundation" field quantization from Origin to any radius specified by the Quantum Virtual Lab/Game technician.

        The next challenge is to verify that the "foundation" field quantization... i.e. coordinate system... supports a Pulsed Emission in which each pulse resolves distribution intelligence/logic required for resolve of QE distribution, in the entire field, on each subsequent pulse.

        Having SIMulated QE Emission within a UQS CAD quantized environment, out to 5 UQS Base Unit Radius Shells, I have found it necessary to develop a computer program to manage the emerging intelligence for subsequent pulsed QE distribution resolve, and am working diligently on it. REF: UQS Virtual Quantum Lab/Game http://www.uqsmatrixmechanix.com/UQSDB.php

        Thanks Hans for your consideration of our model similarities.

        S. Lingo

        UQS Author/Logician

        www.uqsmatrixmechanix.com

        The orthomodular lattice is like classical logic defined by ~25 axioms. It is a skeleton that can get many realizations. Each separable Hilbert space features a set of closed subspaces and this set has exactly the lattice structure of the orthomodular lattice. This the elements of the orthomodular lattices are represented by closed subspaces. Together these subspaces span the whole separable Hilbert space.

        https://en.wikiversity.org/wiki/Hilbert_Book_Model_Project#Relational_structures

        Hi Hans...

        Thanks for the links!!!

        Apparently your model does not emerge from a single Origin Point Source of Energy.

        I am therefor unable to bridge the gap from the Spatial minimum unit of Energy (QE) concept which I derive from the Spherical Origin Singularity of the UQS lattice... i.e. lattice of M4 Orthomodule of type:h=w quantization... to your more complex generalized lattice concepts of Energy.

        However, the more I examine your lattice approach to "Structure in Reality", the more I am convinced that complex generalized concepts of Energy derived from a lattice quantized by an M4 Orthomodule of type: h=w, will eventually bridge to the UQS minimum unit of Energy (QE), and you have developed some "foundational" lattice concepts that may apply to the resolve of minimum quanta of Energy (QE) distribution, as I move outward, pulse by pulse, from the UQS Origin Singularity with my digital Emission SIMulation.

        Thanks for sharing what is obviously a lifetime of work and dedication!!!

        Respectfully,

        S. Lingo

        UQS Author/Logician

        www.uqsmatrixmechanix.com

        Hi Hans van Leunen

        Sue Lingo wrote on Jan. 18, 2018 @ 02:04 GMT said ..."Your initial statement "The name physical reality is used to display the universe with everything that exists and moves therein." is appreciated as a attempt to clarify semantic interpretive errors... but what fundamentally constitutes Physical?... motion??"

        I am thinking of giving "Gravity" as possible answer for the cause of motion ... in your essay "Structure in Reality" dear Hans van Leunen..... Whay do you say?

        By the way I highly appreciate your essay and hope and request you please spend some of your valuable time on Dynamic Universe Model also and give your some of the valuable & esteemed guidance

        Some of the Main foundational points of Dynamic Universe Model :

        -No Isotropy

        -No Homogeneity

        -No Space-time continuum

        -Non-uniform density of matter, universe is lumpy

        -No singularities

        -No collisions between bodies

        -No blackholes

        -No warm holes

        -No Bigbang

        -No repulsion between distant Galaxies

        -Non-empty Universe

        -No imaginary or negative time axis

        -No imaginary X, Y, Z axes

        -No differential and Integral Equations mathematically

        -No General Relativity and Model does not reduce to GR on any condition

        -No Creation of matter like Bigbang or steady-state models

        -No many mini Bigbangs

        -No Missing Mass / Dark matter

        -No Dark energy

        -No Bigbang generated CMB detected

        -No Multi-verses

        Here:

        -Accelerating Expanding universe with 33% Blue shifted Galaxies

        -Newton's Gravitation law works everywhere in the same way

        -All bodies dynamically moving

        -All bodies move in dynamic Equilibrium

        -Closed universe model no light or bodies will go away from universe

        -Single Universe no baby universes

        -Time is linear as observed on earth, moving forward only

        -Independent x,y,z coordinate axes and Time axis no interdependencies between axes..

        -UGF (Universal Gravitational Force) calculated on every point-mass

        -Tensors (Linear) used for giving UNIQUE solutions for each time step

        -Uses everyday physics as achievable by engineering

        -21000 linear equations are used in an Excel sheet

        -Computerized calculations uses 16 decimal digit accuracy

        -Data mining and data warehousing techniques are used for data extraction from large amounts of data.

        - Many predictions of Dynamic Universe Model came true....Have a look at

        http://vaksdynamicuniversemodel.blogspot.in/p/blog-page_15.html

        I request you to please have a look at my essay also, and give some of your esteemed criticism for your information........

        Dynamic Universe Model says that the energy in the form of electromagnetic radiation passing grazingly near any gravitating mass changes its in frequency and finally will convert into neutrinos (mass). We all know that there is no experiment or quest in this direction. Energy conversion happens from mass to energy with the famous E=mC2, the other side of this conversion was not thought off. This is a new fundamental prediction by Dynamic Universe Model, a foundational quest in the area of Astrophysics and Cosmology.

        In accordance with Dynamic Universe Model frequency shift happens on both the sides of spectrum when any electromagnetic radiation passes grazingly near gravitating mass. With this new verification, we will open a new frontier that will unlock a way for formation of the basis for continual Nucleosynthesis (continuous formation of elements) in our Universe. Amount of frequency shift will depend on relative velocity difference. All the papers of author can be downloaded from "http://vaksdynamicuniversemodel.blogspot.in/ "

        I request you to please post your reply in my essay also, so that I can get an intimation that you replied

        Best

        =snp

          Dear Hans,

          Your essay is well written, and I like that you approach the structure in reality. In particular the hierarchy of various mathematical structures one in top of another that you illustrate in the Hilbert Book Model is very close to how mathematicians view the things. Another thing I appreciate is your emphasis on Birkhoff's and von Neumann's quantum logic, and on quaternions, which seem to arise in the most diverse places in physics. I wish you good luck with the contest!

          Best regards,

          Cristi

            Cristi,

            I thank you for your nice comment.

            I do not claim copyright for my documents. So you can use what you want. The pdf files that occur at http://vixra.org/author/j_a_j_van_leunen will also occur as docx files on my website http://www.e-physics.eu .

            Greathings,

            Hans, retired physicist

            Another remark,

            The orthomodular lattice emerges into a structure. That structure can archive dynamic geometric data. However, that structure does not hint at the origin of dynamics and to the fact that this dynamics is rather coherent. It appears that stochastic processes generate the hop landing locations and that the characteristic functions of these processes ensure the coherence of the dynamics. The fact that this characteristic function equals the Fourier transform of the location density distribution of the hop landings indicates that universe is controlled via quaternionic "holomorphic" means.

            In odd dimensions, quaternionic functions feature impulse responses that are shock fronts. The spherical shock fronts integrate into a Green's function.