[UPDATED 02/27/08: I was startled today to find that Jeff Lindsay has written a poem, “Flatland“, inspired in part by this paper. And a wonderful poem it is; go read it.]

I have a half-finished post on the nature of eternity, but I realized that I really need to address some foundational issues with regard to it. And since I co-authored a paper on the subject (“Some thoughts on higher-dimensional realms”, Robert P. Burton and Bruce F. Webster, BYU Studies, Vol. 20, No. 3 (Spring 1980), p. 281 ff.) nearly 30 years ago, I thought I’d simply repost the paper, both in PDF format (as transcribed — not scanned — by BYU Studies) and as text that you can read here.

The gist of the paper is that there is evidence in the scriptures as well as in LDS sources (historical and doctrinal) that there are macroscopic physical dimensions outside of the 3D+time continuum in which we appear to reside. I did this as a University Scholar project my senior year (1977-78) at Brigham Young University; it was the first published writing I ever did (though certainly not the last), and so it’s a bit more awkward and stilted than I would write today. But it still brings together the sources that led me to this conclusion three decades ago, and my studies since then have only reinforced them.

The article itself appears after the jump. I have inserted some comments within brackets ([…]) when appropriate. Footnotes are also indicated with brackets ([1], etc.) and are given at the end of the article. Finally, I have noticed some typographical errors in the PDF transcription of the article (there are sadly quite a few) and have corrected these silently.

Some Thoughts on Higher-dimensional Realms

Robert P. Burton and Bruce F. Webster

[BYU Studies 20, no. 3 (1980)]

For my thoughts are not your thoughts, neither are your ways my ways, saith the Lord.

For as the heavens are higher than the earth, so are my ways higher than your ways, and my thoughts higher than your thoughts. (Isaiah 55:8-9)

In 1975, the Hyperspace Research Group was established at Brigham Young University. The objective of this group has been to develop special computer graphics techniques for generating and presenting meaningful representations of figures with more than three spacial dimensions. [This had to do with research in the BYU Math Department regarding catastrophe theory, which involved hyperdimensional topological shapes.] In the course of research, discussions have often arisen about the likelihood of higher spatial dimensions and how their possible existence might relate to our eternal existence. We were particularly excited by the striking similarities between hypothesized four-dimensional phenomena and certain incidents and descriptions found in the scriptures and other religious literature. The result of our discussion and subsequent investigation form the basis of this article.

Our purpose here is not to prove a theory but rather to propose one by exploring some indications and possibilities that have occurred to us. We have not found our theory to be at odds with accepted doctrine of the Church of Jesus Christ of Latter-day Saints. In no sense do we offer our ideas as prospective new doctrine, but merely for contemplation.

While we have found indications in a variety of locations, we have limited our source material almost exclusively to the standard works of the Church and to statements by Joseph Smith and Brigham Young. There are two reasons for this. First, our readers will primarily be Latter-day Saints, and second, a survey of all discussions and indications of this topic in religious and secular literature would make this article unwieldy and expand its scope at the expense of a closer scrutiny of a rich source of indications. Though we have restricted ourselves primarily to ideas suggested in accepted LDS scriptures and literature, we have been willing to explore some other ideas which, while not precluded by the scriptures, are not dealt with in detail.

There are four major sections in this study. The first is an introduction to the concepts necessary to deal with four (or more) spatial dimensions. Next is a discussion of major indications of higher dimensions found in the scriptures and other LDS sources. The third is an exploration of possibilities of a more speculative nature. Finally, there is a summary with ideas for further investigation.

An Introduction to Hyperspace

Spatial Dimensions

High school geometry has taught us that each new dimension of space builds upon the previous ones. A point, having no dimensions, is moved through space and defines a one-dimensional line segment. This line segment can then be moved in a direction not contained within itself, and the result is a two-dimensional area. If the area is then moved in a direction not contained in itself, it defines three-dimensional volume. Length, area, and volume are all concepts with which we are familiar. But what if we go one step further? What sort of geometric entity do we get if we move a three-dimensional volume in a direction not contained within itself? One might protest that there isn’t such a directions, which for some is an acceptable argument. However, if some choose to limit themselves to three dimensions, we must part company at this point. On the other hand, if the readers are willing to postulate a fourth spatial dimension (and there is no reason not to), then a new realm opens up, one which most can scarcely imagine.

All of our mortal experience has been within a three-dimensional universe, and thus we have no natural conception of a fourth dimension. To permit us to deal with these hard-to-grasp concepts, we will rely upon analogy.[1] We will explore the hypothetical interaction between a two-dimensional world and the three dimensional universe that envelops it. Then, by analogy, we will extend our observations to a three-dimensional universe within a four-dimensional realm.

Flatland Revisited

Let us imagine that we have discovered a two-dimensional world complete with two-dimensional creatures living in two dimensional houses (see Figure 1 on the following page [Sadly, I do not have the illustrations — which were done by Dianne Burton — nor are they in the PDF. However, if you read or review a copy of Flatland, you will get the general idea; here are a few extracted for an article by Ivars Peterson.]). We would find such a world to be strange, indeed, since it and everything in it would be virtually flat.[2] The creatures’ world might be a large flat disk, and the inhabitants thereof would live on its rim. These creatures would also be flat and would probably live in structures consisting of broad lines with gaps in them for entrances and exits.

While observing this novel situation, we would notice that these flat people could perceive and move in only two dimensions! Such beings could not move in a third dimension except through the intervention of an extra-dimensional force and would probably need special help and protection to survive such an experience.[3] If one such 2-D being were to be taken out of his world and then returned, he would be at a loss for words to describe his adventure because he would have nothing in his previous experience to compare it to. His attempts to describe his perceptions to his peers would be met with puzzlement or scorn. He might even be deemed mad.[4]

As 3-D beings, we would quickly see that there would be no areas in two-space hidden from us. All surfaces, including those considered by the 2-D beings to be completely enclosed, would be open to our inspection. Regardless of how many walls lay between a given room and the outside of the house, for example, we would have access to that room as easily as any other. If our 2-D friend were to lock himself in such a room, we could reach him easily and directly. We could even enter the room, but we would meet with one problem: only a two-dimensional cross-section of our body would actually be in that room at any one instant. The “rest” of our body would be on either side of the plane of the two-dimensional world.

We could see not only enclosed areas but also the inside of the walls enclosing the areas. In fact, we could see every particle of every object in the entire 2-D world.[5] No amount of two-dimensional substance could hide anything, because there would be no “behind” from our point of view. The core of the planet would be just as visible as the rocks on the surface, and the skeleton and internal organs of a person as his skin.

Finally, we would notice that, due to the infinitesimal 3-D thickness of this 2-D world, an infinite number of such worlds could be”stacked” on top of one another, permitting worlds without end.[6] These worlds would be closer together than pages in a book, and yet no interaction would occur to them without intervention from a higher-dimensional force. [Note however that some current cosmological theories regarding branes postulate that gravity might interact between such adjacent 3-D realms.] The inhabitants of such a world could never know of the existence of these adjacent worlds unless they were told by someone who had been outside of their own plane of existence.

Extending the Analogy

Our purpose thus far has been to establish some of the basic relationships between an n-dimensional universe and the (n+1) dimensional realm in which it is contained. But now let us suppose that our three-dimensional universe is enclosed in a four-dimensional realm, and that there are beings in that higher realm who are observing us. What would they see? Their initial observation would probably be that our space is “flat” relative to theirs; i.e., it would have only a minuscule extension into the fourth dimension. This earth and the inhabitants thereof (namely us) would all lack extension into the fourth dimension.

They would next observe that, by their standards, we are very limited in our movements and perceptions, functioning only in three dimensions instead of four. If they were to help us “out” of our 3-D predicament, we would find ourselves experiencing perceptions beyond words, or at least beyond familiar words, for our vocabulary is founded upon a three-dimensional experience. If we attempted to describe our adventure to others, we would probably be laughed at or considered insane.

The hyperdimensional beings would quickly discover that there are no volumes or spaces in our dimension into which they could not come. Every room, every container, every vault would be as accessible to them as the 2-D rooms would be to us. A hyperdimensional being — or, rather, a three-dimensional cross-section of such a being — could directly enter a room, explore it, and leave again, without having to pass through doors or windows which might even be locked. There would be no need for the hyperdimensional being to pass through an opening — it would simply exit our three-dimensional universe.

Not only would all spaces be accessible, but every particle of every object would be exposed to view, just as in our observation of the 2-D world. Our skeletons, the earth’s core — all things (and every particle thereof) would be completely visible to these viewers. Nothing would be hidden or obscured, for there would be nothing to hide “behind”.[7]

A last observation they might make is that an infinite number of our three-dimensional universes could be “stacked” on top of one another, as it were, within their four-dimensional realm. These universes would be separated by only an infinitesimal space and yet could have no interaction with each other by their own efforts. The inhabitants of such universes could only know of and influence their “neighbors” through the intervention of some higher-dimensional power.

Summary of Hypothesized 4-D Phenomena

Before proceeding, we summarize the significant characteristics that might accompany the existence of a higher-dimensional realm:

1. A four-dimensional being could access any part of our universe and could enter into and leave any three-dimensional enclosure (e.g., a room) without having to pass through an opening.
2. A four-dimensional view of our three-dimensional universe would allow the viewer to see every particle of our universe.
3. A three-dimensional being would have difficulty describing four-dimensional objects or events, since he would have but a “three-dimensional” vocabulary to call upon.
4. Many three-dimensional universes could be stacked together within a four-dimensional realm. Interaction between such universes could only take place by employing higher-dimensional forces.

We turn now to the scriptures and other prominent LDS literature to look for indications of these hypothesized phenomena.

Major Indications

Comings and Goings

An incident encountered early in our investigations is found in the Gospel According to John. The setting is a gathering of the apostles shortly after Christ’s death:

Then that same evening, being the first day of the week, when the doors were shut where the disciples assembled for fear of the Jews, came Jesus and stood in their midst, and saith unto them, Peace be unto you. (John 20:19)

The point of interest here is that “the doors were shut”, which indicates that the Savior appeared in a closed room. Luke also reports this incident and establishes two important points. First, the disciples were upset by Jesus’ arrival and thought him a spirit. Second, Jesus definitely was not a spirit but had a body of “flesh and bones” (Luke 24:36-39). The sudden appearance of a corporeal body in a closed room could be explained by an entrance from another dimension.

This is not an isolated experience. There is at least one other “closed room” event involving Christ after his resurrection. This occurred during his sojourn with the two disciples traveling to Emmaus (Luke 24:13-32). After stopping to dine with them, we was recognized by the two while he blessed the bread, and then he subsequently disappeared from the room.

Celestial messengers have also made “closed room” appearances. Gabriel appeared to Zacharias inside the temple during an important ceremony to tell him of the forthcoming birth of Zacharias’ son, John the Baptist (Luke 1:11, 19). The angel Moroni visited Joseph Smith’s bedside three times during one night to deliver his message. The account of Moroni’s visit is unique in that it contains a detailed description of the actual coming and going of the angel which might suggest a direct route to another realm:

…I discovered a light appeared in my room, which continued to increase until the room was lighter than at noonday, when immediately a personage appeared at my bedside, standing in the air, for his feet did not touch the floor…[8]

After this communication, I saw the light in the room begin to gather immediately around the person of him who had been speaking to me, and it continued to do so until the room was again left dark, except just around him; when instantly, I saw, as it were, a conduit open right up into heaven, and he ascended till he entirely disappeared, and the room was left as it had been before this heavenly light had made its appearance. (Joseph Smith-History 2:30, 43)

In addition to these appearances, there have been numerous “open area” visitations by the Lord and his angels.[9] These incidents may lack some of the obvious characteristics of a “closed room” experience, but the question of origin and means of arrival still arise when they are carefully considered. For example, where did the angel who appeared to Alma the Younger actually come from (Mosiah 27:11)? And how did he travel here and then return again? The same questions can be asked about Christ’s visit to the Nephites and to the ten tribes, and about the other recorded visits of celestial beings to the earth. A possible answer in accordance with our discussion is not that they traveled some number of light-years from another system through empty space, but that they came quickly and directly from another (and perhaps adjacent) dimension.

There are two other points we wish to explore before leaving this topic. First is the process of translation, or being “caught up into heaven”. [10] In the Book of Moses, we find the account of the removal of an entire city — the city of Enoch — an event which provoked those who remained to exclaim, “Zion is fled!” (Moses 7:69) In the same chapter, we find that those who were converted after the city of Enoch departed were “caught up by the powers of heaven into Zion” to be there also (Moses 7:21, 27). In 2nd Corinthians, Paul describes being “caught up into the third heaven”, possibly while still in a mortal body (2 Cor. 12:2); while in 3rd Nephi, the three disciples of Christ were caught up into heaven and then returned (3 Nephi 28:13-16). All of these quotations refer to a phenomenon in which people and other physical objects are suddenly removed from this planet to another place. Where did they go? How did they get there? Again the question arises: did this involve a traversal of interstellar space, or were the people and things in questions simply moved to an adjacent realm? For the authors of this article, the inconveniences of interstellar travel, even at speeds near the speed of light, make the latter explanation more acceptable.

The second topic deals with the creation of the earth. Brigham Young, speaking of the Creation and the Fall, used some very unusual language when referring to where the earth came from at the time of the Fall and where it will return someday:

When the earth was framed and brought into existence and man placed upon it, it was near the throne of our Father in heaven. And when man fell…the earth fell into space, and took up its abode in this planetary system, and the sun became our light….This is the glory the earth came from, and when it is glorified it will return unto the presence of the Father, and it will dwell there, and these intelligence beings that I am looking at, if they live worthy of it, will dwell upon this earth.[11]

The description of “falling into space” and then leaving to return to the former place suggests a realm “outside” our universe. Once gain, this can be understood in the context of a fourth spatial dimension.

Four-dimensional Perceptions

Another indication of a possible existence of a fourth spatial dimension comes from the descriptions found in prophetic visions. An example is the vision of Moses:

…Moses cast his eyes and beheld the earth, yea, even all of of it; and there was not a particle which he did not behold, discerning it by the spirit of God.

And he beheld also the inhabitants thereof, and there was not a soul which he beheld not; and he discerned them by the Spirit of God; and their numbers were great, even numberless as the sand upon the sea shore. (Moses 1:27-29)

Moses’ comment that there was “not a particle” that he did not see parallels exactly our hypothesis that someone viewing our world from the fourth dimension would see every particle thereof.

A more unusual vision is described in the first chapter of Ezekiel:

And I looked, and behold, a whirlwind came out of the north, a great cloud, and a fire infolding itself, and brightness was about it….

Also out of the midst thereof came the likeness of four living creatures. And this was there appearance; they had the likeness of a man.

And every one had four faces, and every one had four wings….

And they had the hands of a man under their wings on their four sides; and they four had their faces and their wings.

Their wings were joined one to another; they turned not when they went; they went every one straight forward.

…whither the spirit was to go, they went; and they turned not when they went…

And the living creatures ran and returned as the appearance of a flash of lightning.

Now as I beheld the living creatures, behold one wheel upon the earth by the living creatures….

The appearance of the wheels…was as it were a wheel in the middle of a wheel.

Where they went, they went upon their four sides; and they turned not when they went. (Ezekiel 1:4-18)

The highly unusual and somewhat confusing language that Ezekiel used to describe what he saw might well be the result of the inadequacy of our language to describe hyperdimensional objects or events. His description of a a “wheel in the middle of a wheel” could suggest a four-dimensional torus, just as a hypercube is often represented as a cube within a cube (see Figure 2 [here are some representations of a hypercube; scroll down a bit]). The reference to multiple-sided beings who went straight forward (though, apparently, facing in different directions) parallels part of our description of a hyperobject entering our space.[12] These, of course, are only possibilities, but we feel that they may be viable.

Earlier, we alluded to the experience of the three Nephite disciples being caught up into heaven. What they saw there they were forbidden to discuss. The description of this event, however, carried the implication that their experience simply could not be articulated:

And behold, the heavens were opened, and they were caught up into heaven, and saw and heard unspeakable things.

And it was forbidden them that they should utter; neither was it given unto them power that they could utter the things which they saw and heard. (3 Nephi 28:13-14)

Adjacent 3-D Universes

The first possibility that springs to mind when considering the topic of adjacent 3-D universes is the nature and location of that realm which we call the “spirit world”:

The spirits of the just are exalted to a greater and more glorious work; hence they are blessed in their departure to the world of spirits. Enveloped in flaming fire, they are not far from us, and know and understand our thoughts, feelings, and motions, and are often pained therewith.

Flesh and blood cannot go there; but flesh and bones, quickened by the Spirit of God, can.[14]

There are myriads of disembodied evil spirits — those who have long ago laid down their bodies here and in the regions round about among and around us; and they are trying to make us and our children sick, and are trying to destroy us and tempt us to evil.[15]

When you lay down this tabernacle, where are you going? In to the spirtual world….Where is the spirit world? It is right here.[16]

The preceding quotations of Joseph Smith and Brigham Young indicate that there is a world of spirits very near to us and yet seldom perceived by us. While those spirits can view us and can to a certain degree affect us, we cannot see them without special help from God. Enoch, for example, anointed his eyes as directed by the Lord and was able to see spirits and other “things which were not visible to the natural eye” (Moses 6:35-36). Elisha prayed, and the servant of Elisha beheld the army of spirits surrounding the besiegers of Dothan:

And Elisha prayed, and said, Lord, I pray thee, open his eyes, that he may see. And the Lord opened the eyes of the young man; and he saw; and, behold, the mountain was full of horses and chariots of fire round about Elisha. (2 Kings 6:17)

Other indications of the existence of adjacent 3-D universes are found in the books of Moses and Abraham. In the vision referred to earlier, Moses saw this:

And he beheld many lands; and each land was called earth, and there were inhabitants on the face thereof.[17] (Moses 1:29)

This passage could be interpreted at least three ways. First, Moses could have seen other planets within our own universe; second, he could have seen different planets adjacent to the earth in different 3-D universes; or third, he could have seen our earth at several different points in time. An echo of the image of multiple worlds is found in Abraham’s vision:

And he said unto me…,behold, I will show you all these. And he put his hand upon mine eyes, and I saw those things which his hands had made, which were many; and they multiplied before mine eyes, and I could not see the end thereof. (Abraham 3:12)

Brigham Young has also indicated that there are more kingdoms than we realize:

How many kingdoms of glory there are, I know not; and how many degrees of glory there are in these kingdoms, I know not; but there are multitudes of them.[18]

The kingdoms that God has prepared are innumerable.[19]

Additional Possibilities

Our purpose in this section has been to present ideas that have come as we have pondered and discussed the implications of a possible hyperdimensional existence. Simply put, we are taking a fresh look at certain gospel teachings in light of the hypothesis that there is a hyperdimensional existence, to see if the hypothesis may be consistent with these teachings and perhaps of some value in helping us to comprehend some heretofore difficult-to-understand occurrences.

Increasing Dimensions

After one has entertained the possibility of a fourth spatial dimension, the next question almost inevitably is — Is there a fifth? A sixth? Then, is there an end to it? If there might be additional dimensions, then interesting interpretations of two basic doctrines — different kingdoms of glory and eternal progression — invite consideration.

Kingdoms of Glory. The seventy-sixth section of the Doctrine and Covenants tells us that there are three principal kingdoms or realms of glory: the celestial, the terrestrial, and the telestial (vv. 50-113). Those in the celestial kingdom will enjoy the presence of God and Christ (v. 62); those in the terrestrial will have just the presence of Christ (v. 77); and those in the telestial, the presence of neither (vv. 86, 112). The idea that the inhabitants of a particular kingdom cannot visit higher kingdoms but can visit lower kingdoms is implicitly (and to a degree explicitly) stated in this section (vv. 77, 86-88, 112). One possible explanation for this limitation would be that each kingdom has one more spatial dimension than the next lower kingdom. Thus the inhabitants of a given kingdom could not enter into the next higher kingdom any more than we could by our own power leave this three-dimensional space and move about in one which ahs four dimensions. The inhabitants of a dimensionally-superior kingdom could, however, visit any lower kingdom contained within their space, just as we could visit a two-dimensional world if such a world existed within our universe. This brings us to our next point.

Eternal Progression. We can consider the concept of eternal progression in a similar fashion. Joseph Smith’s statement about progression and exaltation takes on a special meaning if our hypotheses are correct:

What did Jesus do? Why, I do the things I saw my Father do when worlds came rolling into existence. My Father worked out his kingdom with fear and trembling, and I must do the same; and when I get my kingdom, I shall present it to my Father, so that he may obtain kingdom upon kingdom, and it will exalt him in glory. He will then take a higher exaltation, and I will take his place, and thereby become exalted myself. So that Jesus treads int he tracks of his Father and inherits what God did before; and God is thus glorified and exalted in the salvation and exaltation of all His children.[20]

We wonder if that advancement in exaltations might not be advancement to a higher dimension. This idea has at least two implications. First, it would give an exalted person dominion over an infinity of dimensionally inferior realms similar to the realm of his previous dominion. Second, it would allow each of those “under” him to advance to the position previously occupied by him. In light of this possibility, the following passages from the Doctrine and Covenants offer a new meaning:

All kingdoms have a law given;

And there are many kingdoms; for there is no space in the which there is no kingdom; and there is no kingdom in the which there is no space, either a greater or a lesser kingdom.

And unto every kingdom is given a law; and unto every law there are certain bounds also and conditions. (D&C 88:36-38)

This earth, in its sanctified and immortal state, will be made like unto crystal and will be a Urim and Thummin to the inhabitants who dwell thereon, whereby all things pertaining to an inferior kingdom, or all kingdoms of a lower order, will be made manifest to those who dwell on it; and this earth will be Christ’s.

Then the white stone mentioned in Revelation 2:17 will become a Urim and Thummin to each individual who receives one, whereby things pertaining to a higher order of kingdoms will be made known. (D&C 130:9-10)

Here, explicit reference is made to greater (or higher) and lesser kingdoms, filling up all of existence. In fact, the last passage quote seems to indicate that there will be an order of kingdoms higher than the celestial kingdom that will exist on this earth after its sanctification. Again, Brigham Young is not without comment on this subject:

And when we have passed into the sphere where Joseph [Smith] is, there is still another department, and another, and another, and so on to an eternal progression in exaltation and eternal lives.[21]

Nature of Time

The nature of time, a subject far too complex to be given just treatment here, still deserves mention. In various places in the scriptures, we read where prophets were shown the future with detailed accuracy, such visions often encompassing the entire history of the earth.[22] For example, Joseph Smith said:

The great Jehovah contemplated the whole of the events connected with the earth, pertaining to the plan of salvation, before it rolled into existence, or ever “the morning stars sang together” for joy; the past, the present, and the future were and are, with him, one eternal “now”.[23]

Because of this quote as well as passages in the scriptures and elsewhere, we offer the possibility that time as we perceive it may be itself a dimension, and that those existing in higher dimensions may view time in its entirety (see Alma 40:8), just as one could see the entire Grand Canyon from the air but could only see a small portion at a time if traversing it by foot. We offer this idea as a suggestion and forgo its exploration at present.

Conclusion

Lest any of our readers misunderstand the nature and purpose of this article, we emphasize the following:

1. We do not claim to have proved or established anything, nor do we think that our ideas are provable at this time. We are only exploring possible (though, to us, somewhat satisfying) explanations of certain passages and statements by major LDS authorities.
2. We do not claim that our explanations of these passages are correct — merely possible.
3. We do not claim that the concept of hyperspace resolves anything, though we do feel that it both simplifies concepts and forms an apparently consistent framework in which to contemplate.
4. Though we have not restricted ourselves exclusively to accepted ideas, we do not find our comments to be at odds with LDS doctrine or teachings; rather, our speculations offer a more comfortable explanation of some heretofore difficult-to-understand concepts and occurrences.

The purpose of this paper has been to consider the possibility of hyperdimensional realms, using scriptures and statements of prominent LDS authorities as principal sources. We invite further discussion on the subject. We suggest these additional areas of exploration:

1. A new survey of religious and philosophical books written on the subject.
2. A study of religious language intimating hyperdimensional references, such as the passage from Isaiah.
3. A study of the nature of man including birth, body-spirit relationships, and near-death and clinical-death descriptions from LDS literature and other sources.
4. An analysis of indications found in such sources as apocryphal literature, literature from other religions, and comments from other LDS authorities.
5. Some speculation on the physics of the situation. For example, a very small 4-D thickness might become significant on a subatomic level [shades of string theory!]. How might that affect particle behavior?
6. Some contemplation on how the idea of higher dimensions might affect our perceptions of ourselves, of our relationship with God, and of our understanding of the universe.

Investigation of these and other related topics may lead us to a greater understanding of the nature of existence and our position therein. We hope that such an understanding will humble us through a tiny glimpse of the glory of God and will help us to draw closer to our Eternal Father, to follow in his footsteps forever.

When I consider thy heavens, the work of thy fingers, the moon and the stars which thou hast ordained; what is man that thou are mindful of him? and the son of man, that thou visitest him? For thou hast made him a little lower than the angels… (Psalms 8:3-5)

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[from 1980:]

Robert P. Burton, associate professor of computer science, is head of the Hyperspace Research Group.

Bruce F. Webster, at the time he co-authored this article as a University Scholar Project, was a member of the Hyperspace Research Group.

====================================

[1] Edwin A. Abbott, Flatland, a Romance of Many Dimensions, 5th ed. rev. (Oxford: Basil Blackwell, 1944). See also Claude Bragdon, A Primer of Hyperspace (London: Andrew Dakers Limited, 1939); G. Hodson and A. Horn, Some Experiments in Four Dimensional Vision (Plymouth, Mass: William Brendon and Son, Ltd., 1933); M. Kendall, A Course in the Geometry of n Dimensions (London: Charles Griffin and Company, Ltd., 1961); and C. Lindgren and S. Slaby, Four-Dimensional Descriptive Geometry (New York: McGraw-Hill Book Company, 1968).

[2] There must be some three-dimensional thickness or else there can be no interaction at all between a three-dimensional object and the two-dimensional world, since no actual cross-section of the three-dimensional object could be defined.

[3] Since such a being would be infinitesimally thin by our standards, he would probably fall apart upon being subjected to three-dimensional forces.

[4] Such was the fate of the protagonist of Edwin Abbott’s classic work, Flatland. He could only come up with the phrase, “Up, but not North!”

[5] Ignoring, of course, the problem of distance.

[6] Assuming our space is infinite. If it is not, then the number is still very great, but not infinite.

[7] Ignoring again the problem of distance.

[8] A higher-dimension personage would not need to stand upon the floor; there would probably be no gravitational attraction between him and the elements of our three-dimensional world.

[9] See Joseph Smith-History 2:16-17; 3 Nephi 11:8, 27:1-3; Mosiah 27:11; Daniel 10:5-7; 1 Corinthians 15:5-8; etc.

[10] While these two phenomena may not be identical, they are similar enough to warrant consideration together.

[11] Journal of Discourses, 26 vols. (London: Latter-day Saints’ Book Depot, 1855-1886), 17:143; hereafter cited as JD. See also JD, 9:317.

[12] Abbot, Flatland, p. 72.

[13] See also 3 Nephi 17:15-17.

[14] Joseph Smith, Teachings of the Prophet Joseph Smith, comp. Joseph Fielding Smith (Salt Lake City: Deseret Book Co., 1938), p. 326.

[15] JD, 6:73; see also JD, 5:54-55.

[16] JD, 3:369.

[17] Moses 1:11 explains that one must have spiritual eyes to see the things of God.

[18] JD, 9:107.

[19] JD, 8:154.

[20] Smith, Teachings, p. 347-48.

[21] JD, 3:375.

[22] Moses 1 and 7; Ether 3:25-26; Mormon 8:35; Isaiah 29; etc.

[23] Smith, Teachings, p. 220; see also D&C 84:100.