1. I am skeptical of the advisability or feasibility of using scientific deductions about the natural realm to show or demonstrate the Triune nature of God. My skepticism is rooted in the nature of God as being sui generis—completely unique. There is nothing like him. His being is of a qualitatively different kind. There is an unbridgeable Creator/creature distinction. This is why his trinal nature is utterly unique and no natural analogies suffice to adequately capture the Trinity.
Having mentioned my skepticism I do recognize that theologians have looked for what are called vestigia Trinitatis—“the drawing of analogies to the Trinity from the threefold related structure of certain created things.” This use of analogies must be done carefully. Some analogies of the Trinity that have been developed actually better teach any one of the number of Trinitarian heresies (i.e., tritheism, modalism, or some variant of subordinationism). We should always recognize that even the best analogies cannot fully capture the infinite nature of the Triune God. Physics professor Michael J. Bozack, in an interesting article entitled “Physics in the Theological Seminary, quotes Ian Barbour on the nature of “analogy”:
Analogy is the extension of patterns of relationship drawn from one arena of experience to coordinate other types of experience…Analogy is never a total identity or a comprehensive description, but only a simplified comparison of limited aspects.
That last phrase—“simplified comparison of limited aspects”—is crucial. By remembering this and articulating this for your hearers I believe your presentation may be powerful as a conceptualization tool.
The Trinity is hard to understand because it is so unlike most of our experience. We are accustomed to persons and beings being nicely correlated one-to-one. In the Trinity we have the one eternal being of God existing in three eternal persons—Father, Son, and Holy Spirit. The three-in-one relationship can be analogized to some degree but we must always remember the limitations of such analogies. For example, your discussion about time, space, and matter being dimensions and not simply ingredients like the triad of the peanut butter and jelly sandwich was good. This does show an inseparable three-ness but more is needed to properly image the Trinity. The nature of the dimensions of space, time, and matter is not such that any individual dimension can be said to fully instantiate the universe. In the Trinity, however, the fullness of divine being is fully contained in each person. Louis Berkhof states it this way:
The whole undivided essence of God belongs equally to each of the three persons. This means that the divine essence is not divided among the three persons, but is wholly with all its perfections in each one of the persons, so that they have a numerical unity of essence.
From this Berkhof notes, “It is especially when we reflect on the relation of the three persons to the divine essence that all analogies fail us and we become deeply conscious of the fact that the Trinity is a mystery far beyond our comprehension.” Thus, all your analogies fail to capture this dynamic of the Trinity. The example of height, width, depth does show a three-ness but “height” by itself does not contain the fullness of space within it. On the other hand, Jesus Christ does contain all the fullness of God within him (Colossians 2.9).
I would not attempt to use science to demonstrate the Trinity. The Trinity is a concept specifically revealed in the coming of Christ and the coming of the Holy Spirit in the history of redemption. I can’t resist quoting Benjamin B. Warfield at length because he so wonderfully articulates this idea:
The simplicity and assurance with which the New Testament writers speak of God as a Trinity have, however, a further implication. If they betray no sense of novelty in so speaking of Him, this is undoubtedly in part because it was no longer a novelty so to speak of Him. It is clear, in other words, that, as we read the New Testament, we are not witnessing the birth of a new conception of God. What we meet with in its pages is a firmly established conception of God underlying and giving its tone to the whole fabric. It is not in a text here and there that the New Testament bears its testimony to the doctrine of the Trinity. The whole book is Trinitarian to the core; all its teaching is built on the assumption of the Trinity; and its allusions to the Trinity are frequent, cursory, easy and confident. It is with a view to the cursoriness of the allusions to it in the New Testament that it has been remarked that "the doctrine of the Trinity is not so much heard as overheard in the statements of Scripture." It would be more exact to say that it is not so much inculcated as presupposed. The doctrine of the Trinity does not appear in the New Testament in the making, but as already made. It takes its place in its pages, as Gunkel phrases it, with an air almost of complaint, already "in full completeness" (vollig fertig), leaving no trace of its growth. "There is nothing more wonderful in the history of human thought," says Sanday, with his eye on the appearance of the doctrine of the Trinity in the New Testament, "than the silent and imperceptible way in which this doctrine, to us so difficult, took its place without struggle - and without controversy - among accepted Christian truths." The explanation of this remarkable phenomenon is, however, simple. Our New Testament is not a record of the development of the doctrine or of its assimilation. It everywhere presupposes the doctrine as the fixed possession of the Christian community; and the process by which it became the possession of the Christian community lies behind the New Testament.
We cannot speak of the doctrine of the Trinity, therefore, if we study exactness of speech, as revealed in the New Testament, any more than we can speak of it as revealed in the Old Testament. The Old Testament was written before its revelation; the New Testament after it. The revelation itself was made not in word but in deed. It was made in the incarnation of God the Son, and the outpouring of God the Holy Spirit. The relation of the two Testaments to this revelation is in the one case that of preparation for it, and in the other that of product of it. The revelation itself is embodied just in Christ and the Holy Spirit. This is as much as to say that the revelation of the Trinity was incidental to, and the inevitable effect of, the accomplishment of redemption. It was in the coming of the Son of God in the likeness of sinful flesh to offer Himself a sacrifice for sin; and in the coming of the Holy Spirit to convict the world of sin, of righteousness and of judgment, that the Trinity of Persons in the Unity of the Godhead was once for all revealed to men. Those who knew God the Father, who loved them and gave His own Son to die for them; and the Lord Jesus Christ, who loved them and delivered Himself up an offering and sacrifice for them; and the Spirit of Grace, who loved them and dwelt within them a power not themselves, making for righteousness, knew the Triune God and could not think or speak of God otherwise than as triune. The doctrine of the Trinity, in other words, is simply the modification wrought in the conception of the one only God by His complete revelation of Himself in the redemptive process. It necessarily waited, therefore, upon the completion of the redemptive process for its revelation, and its revelation, as necessarily, lay complete in the redemptive process.
Warfield’s full discussion is well worth reading in its entirety.
One other danger of tying our theology too closely to modern science is that the theories and findings of science can change rather quickly. It is has been said that when theology marries science, theology will soon be a widow. The other facet to be considered is that scientific findings are open to multiple metaphysical interpretations. There are some who look to the recent discovery of the Higgs Boson as confirming that the divine force may be a Pentany (a five-fold conception of deity) and not a Trinity! Here a few comments I found online:
The infamous “God particle”, discovered in the Hadron Collider, suggests that the Godhead is a Pentany, not a Trinity!
The Hadron particle collider may be paying off, finally, with some surprisingly practical advice. Latest experiments indicate that there are not just one Higgs particle, but five, each with a similar amount of mass, but with different electrical charges. These appear to be balanced, also, with one positive face, one negative face, and three neutral faces. This was apparently demonstrated by “splitting out” the long sought after Higgs / Boson particle.
This arrangement of five forces appears like the perfect expression of Pentany Theory which describes the Godhead as consisting of the Father -as positive Yang, the Divine Mother -as negative Yin, with the Son, Daughter and Great Spirit impulses as neutral because they are essentially “metaphorical children” of the Alpha and Omega forces.
Now I think this guy is completely wrong ... but this does demonstrate something important. Scientific data is screened through our prior worldview presuppositions. Scientific facts do not come to us in some neutral fashion. We bring an interpretative framework to the facts. Worldview formation is a multi-discipline affair in which we bring revelation, theological, historical, scientific and philosophical perspectives to bear on developing a consistent and coherent worldview. Science does not have a privileged place in worldview formation.
2. I mentioned above that by some re-framing of the presentation the conceptual issue might have some force. Remembering Barbour’s words from above—“simplified comparison of limited aspects”— I think your presentation could be useful if you make a comparison between the complexity and seeming counter-intuitive nature of both the Trinity and elements of quantum theory. Many times the doctrine of the Trinity is belittled because of its complexity. Some even argue for logical contradiction. Now there is no formal contradiction in the orthodox understanding but there are elements that go beyond our full comprehension. We want to know “how” it can be the case that one infinite being can exist in three eternal persons. There doesn’t seem to be anyway to answer this “how” question. By utilizing some of the details of modern quantum theory we seek, not to prove the Trinity, but, rather, to demonstrate that even in the arena of the created realm there are complexities and counter-intuitive realities that nevertheless exist (i.e., the particle/wave functions of light). This is where I would see reflection upon God’s amazing universe helping the Christian.
**In the process of further correspondence I referenced the following:
I mentioned an essay by Michael J. Bozack—“Physics in the Theological Seminary.” I read an essay recently that mentioned his work. Robert Clifton and Marilyn Regehr interact with an essay by Bozack in which compares the God-man unity of Christ with wave-particle duality. In responding to Bozack they write the following which I think may be very relevant to your presentation:Bozack’s purpose, though not stated in the clearest terms, is to use wave-particle complementary as an aid to our understanding, so often rooted in preconceived, deep-seated views that such a thing as the hypostatic union seems impossible. Taken as a metaphor for understanding rather than as a validation of a religious concept by an appeal to science, this approach is unproblematic. Yet, in our opinion, it can lead to danger because it invites the reader to “read between the lines.” The parallels look so remarkable that the theological concept under examination not only gains clarification in the eyes of the reader, but its apparent truth appears to receive confirmation from the scientific realm. The implication is that because it is possible to think a certain way in science, it is legitimate to think the same way within theology. Perhaps this is the reader’s problem. Yet, when taking this approach, it is the author’s responsibility to clearly delimit his purpose.
Their distinction between a metaphor versus a validation of a religious concept is important. This is another way of describing what I was attempting to get at with my [earlier comments].
 Robert K. Clifton and Marilyn G. Regehr, “Toward a Sound Perspective on Modern Physics: Capra’s Popularization of Mysticism and Theological Approaches Reexamined” Zygon 23 (1990), 92. They are interacting with Bozack’s article “Conjugate Properties and the Hypostatic Union,” Perspectives on Science and Christian Faith: Journal of the American Scientific Affiliation 39 (1987), 105-107. Robert K. Clifton and Marilyn G. Regehr, “Toward a Sound Perspective on Modern Physics,” 92.
 Stanley J. Grenz, David Guretzki, and Cherith Fee Nordling, Pocket Dictionary of Theological Terms (Downers Grove: Intervarsity Press, 1999), 119. For a brief discussion see John Frame’s “Trinitarian Analogies” available at: http://www.frame-poythress.org/trinitarian-analogies/.
 Michael J. Bozack, “Physics in the Theological Seminary” JETS 36 (1993): 1993. Bozack’s article is available here: http://www.etsjets.org/files/JETS-PDFs/36/36-1/JETS_36-1_065-076_Bozack.pdf.
 Benjamin B. Warfield, “The Biblical Doctrine of the Trinity” in The Works of Benjamin B. Warfield, vol. 2, (Grand Rapids: Baker, 2000 [Original: Oxford University Press, 1932]), 143-144. Available online here: http://www.monergism.com/thethreshold/sdg/warfield/warfield_trinity.html.
 http://2012forum.com/forum/viewtopic.php?f=6&t=19085. I believe this person was referring to the fact that some scientists theorize there may be up to five bosons and not simply one. Here is an article on this that contains more bibliographic data: “Physicists Indicate Existence of Multiple Higgs Bosons” Dartmouth Undergraduate Journal of Science (Spring, 2013), n. p. Online: http://dujs.dartmouth.edu/physical_sciences/physicists-indicate-existence-of-multiple-higgs-bosons#.Um3ftyj_Tao.