The following sections discuss various ideas connected with the fourth spatial dimension. The last three categories in the list below follow the sequence presented in the Introduction

1. Astronomy/Cosmology Links. Links of current interest in astronomy and cosmology are given here.

2. General Links (Physics, Math). Several mathematics and physics links appear here.

3. Other Links (Art, Architecture, History). Not only are mathematicians, physicists, and cosmologists interested in the fourth spatial dimension, so are several artists and some architects. There are a few links here which show that there is interest in extra dimensions outside the realm of science and mathematics. For example. the best history of the fourth spatial dimension in the early twentieth century was written by an art professor, Linda D. Henderson, rather than a cosmologist, and so a link to her book appears in this section.

4. Links on a "Small" Fourth Spatial Dimension (String theory). These links describe the fourth spatial dimension as it is understood in string theory.

5. Links on a "Large" Fourth Spatial Dimension (Hyperspheres). This is where I park any link which mentions a structure with topological positive curvature consistent with (or similar to) Riemann's hypersphere. If the discussion infers that a spherical aurface is "embedded" in higher dimensional space, it goes here, as does one or two references to "hypercubes" (or "tesseracts")

6. Links on a "Medium" Fourth Spatial Dimension (Gravity). Many physicists theorize that a (tiny) fraction of gravity might occur in space involving a fourth spatial dimension. This is the "hybrid" variation: the fourth spatial dimension here is assumed to be neither as large as the way is referred to in Riemann's hypersphere, nor is it as small as the way it is in string theory. The size is estimated in the first link (though not all physicists agree on the size of that extra dimension).

A. Astronomy/Cosmology Links

There's a slight bias here in that a few of these links refer to the possibility of closed universe (which is what the FRW metric is). These files were originally at the start of Essay #3, right after I presented the most important parameters of the universe. On second thought, it makes more sense to place them here.

1. Cosmology File by Ned Wright. This is without doubt the best site for cosmology on the WEB, and I've been saying this for years (even though Dr. Wright is convinced that space is "flat" and tends to exclude references supporting "non-flat" models). There are two types of "news" files - one with current news and the other an archive of "old news". Right now (April, 2005), the only the first item below is in the "new news" file; everything else is in the "old news" file. Assuming that you get to the right file, you'll probably have to scroll down to get most of what follows at Dr. Wright's site:

   1. 11 Jan 2005: Cosmic Ripples Seen by Galaxy Surveys. This article mentions two surveys recently completed - Sloan digital Sky Survey (SDSS) and the 2 Degree Field Galaxy Redshift Survey (2DF) - and that the combined data seem to suggest a universe that's barely closed.

   2. 9 October 2003: Take a look at "A twelve-sided universe? Probably not?" (Since this is back in October 2003, you'll have to scroll toward the bottom of the file.) The idea that the universe might be some sort of many sided crystal surfaces from time to time, and the latest data just might be able to prove it. While Luminet at al. propose this possibility in Nature, Cornish et al. unambiguously say "no way!" in an article submitted to Physics Review Letters.

   3. 1 May 2003: The discussion and all links in "Many, many supernovae" are worth a look. The modest surprise is that farther supernovae seem to be brighter than expected. The farthest supernova at z = 1.755 is discussed in Essay #6 on anomalies.

   4. 11 Feb 2003: Dr. Wright discusses the results of the Multiple Anistropy Probe (MAP) and reiterates the conclusion that the universe seems flat. Since this is the most recent and presumably the most accurate data, then some of the references below suggesting that the universe might be closed are probably less persuasive, although I have chosen to include them anyway.

   5. 9 October 2002: A balloon experiment called ARCHEOPS indicates that the universe is "flat" with no more than a 2% to 3% error either way. If you take the ARCHEOPS link, you'll see a graph which compares this result to other measurements. A similar graph appears in Item d. below.

   6. 24 April 2002: (Scroll down from Item b. above.) Note the deduction of the age of the universe at 13 to 14 billion years by studying how old the oldest white dwarfs are in M4 (a Messier globular cluster about 7,000 light years away, very easy to find with binoculars 1º west of Antares). This age agrees nicely with several other techniques mentioned shortly.

   7. 29 April 2001: At considerable effort, Dr. Wright keeps the charts up to date with the latest findings of a variety of probes into the very early universe. What is particularly intriguing is the noticeable change in the data points between 2000 and 2001. Several links below provide the cosmological interpretation of the chart data.

2. The NASA Astrophysics Data System (ADS) is a valuable tool to research a base of over 1,000,000 abstracts, though the actual articles are usually not available unless they is posted at the authors' home pages. Here is the query form, and if you plug in key phrases like "extradimensional cosmology", "Riemannian manifolds", "FRW metric", "fourth spatial dimension", etc., into the field of "abstract words", you'll get 100 of the most recent abstracts with those key words. One of the things you're sure to notice is the recent surge of interest in extra spatial dimensions and non-conventional cosmology.

   Whenever there is a reference to an abstract in these essays, over 90% of the time I got it using this query form.

3. This link to the Wilkinson Multiple Anisotropy Probe (WMAP) provides the latest (May 2005) estimates of the global parameters of the universe. These are the ones most commonly quoted these days.
4. Quintessence appeared in the November 2000 issue of Physics World and was written by Robert R. Caldwell and Paul J. Steinhardt. This is a fine "survey" articles and tells why the assumption these days is a universe with about one-third dark matter and two-thirds dark energy (a very good guess, according to WMAP in the previous link). The text near Figure 3 suggests a no-doubt-about-it flat universe, but other data suggests there might be room for a little doubt.

Several of the links following the article are worth a look. One I particularly like is the first link (SNAP.lbl.gov) in the section entitled "Evidence for an Accelerating Universe". If you take the "News" link on the first display, you'll get several intriguing articles concerning the findings.

5. The Dark Side of the Universe is an article in the March 8, 2002 Christian Science Monitor written by Michelle Thaller. (Fortunately, the CSM keeps its archived articles available for a long time.) It puts the latest results of the microwave background analysis in good perspective. If there's any hubris these days, it comes in knowing for sure that we don't know very much at all. For example, the discovery of "dark energy" in recent years has esurrected Einstein's "cosmological constant", once regarded by him as his biggest mistake. How many other surprises are there?

6. Although it was presented in the Introduction, this article in 1932 shows when Einstein changed his allegiance from a positively curved model (with Riemannian space) to a more conventional Euclidean model (with "flat" space). Scroll down until you get to "Einstein and De Sitter Return to Euclidean Idea of Cosmos". The popularity of the Einstein - de Sitter universe is probably why interest in extra-dimensional space was waning at this time.

   In my opinion, here are the two most intriguing passages in the article about Einstein - de Sitter's Euclidean model:

   A ray of light would not traverse the circuit of the universe and come back to where it started as it would in the superseded Einstein and other varieties of space. Curvature of space is on the average banished from the universe......The geometry of an Einstein universe is based on the assumption that light travels in straight lines.

   What I'll do here is demonstrate that it's possible, well within the confines of "big bang theory", to define the universe when t = 0 - that is, the singularity - in such a way as to show that light does traverse a circuit of the universe and does come back to where it started. That definition gives us a universe in which light does not travel in straight lines.

B. General Links

1. Tetraspace by Garrett Jones is a relative newcomer to the field. It came "on-line" in 2001 and I became aware of it in late 2003. Compared to those at my site, you'll find common links and quite a few other useful ones. In fact, I think he has some of the most interesting links on the web. And, if you're a "blogger", both this site and Eric Saltsman's site (below) have discussion forums.

2. Cristina Sormani maintains a home page which has many good links on Riemannian space. If you take the link to "What is Riemannian geometry? A description for the non-mathematician", you'll see a similar treatment of the topic. One of the things I've noticed is that the mathematicians concentrate (understandably) on the geometric/mathematical aspects of this space, and rarely mention how the curvature of such space would affect our perception of discrete, luminous objects in it. In Essay #2, you'll see how Riemannian space - which I call "hyperspace" - will affect our observations, and later essays will ask whether or not a variety of unusual observations in deep space give any indication that the universe actually has Riemannian space.

3. Josh Erlich has written a a very short file five-page pdf file about the history about the fourth spatial dimension. At the end of the pdf file is a link to the best short, yet comprehensive, history I've come across on the fourth spatial dimension: Why bother with Extra Dimensions? This is the abbreviated PhD thesis of Sanjeev Seahra. The last file - "The fifth dimension at the turn of the current century" - explains why there is renewed interest in a large fourth spatial dimension.

4. The French seem to be at the forefront of alternate cosmologies, and one of the leaders there is Jean-Pierre Luminet. This link is to a file where he lists his previous publications, and near the top are links to two good survey articles:: Is Space Finite and Topology of the Universe. Although the "dodecahedral universe" has not yet met with much acceptance - Ned Wright panned it in his "News" section - the reference review by George Ellis is worth a look.

   [Here's something you might find interesting. Near the start of Dr. Luminet's short essay on the topology of the universe, he quotes Archytas. That same reference is about two-thirds of the way down Archytas' biography file. I think you will agree that the quote can be interpreted equally well as favoring either (1) an infinite universe and/or (2) a universe with no boundary.

   Aristotle was less inclined to believe in the notion of "infinite", as you will see as you scroll down to the end of this file. The cosmological model in vogue at the time assumed the earth was at the center of the universe, and that everything moved around the earth in crystalline spheres, including the stars. Aristotle did not think the stars were at an infinite distance, because that would imply an infinite velocity around the earth, which ran counter to his intuition.

   Although Aristotle disagreed with Archytas' notion about infinity, by combining selected components of their quite reasonable deductions, the two of them proposed the first broad outline of a "finite but unbounded" universe. The boundary problem has always been a concern, and apparently was why Riemann proposed the hypersphere back in 1851, as mentioned in the article Is Space Finite?]

5. Flatland by Edwin Abbott. This is the classic in the field, a popular book in college courses, and a "must read" (not to mention a good read as well).

   I'm going to do you a favor: here's the Project Gutenberg Search Page. Simply insert "Edwin Abbott" as the author or "Flatland" the title in the Search Area, and you'll get two titles in the Gutenberg files - one illustrated, one not. (The first time you activate the "search" icon, you'll probably be routed to the nearest of several search sites, and will have to type in the author/title again.) On the same page you might want to click on the "Browse" link as it will lead you to a file when you can search the book inventory alphabetically by author or title. The objective of the Project Gutenberg is to add classic books (whose copyright has expired) that can be downloaded for the user's convenience.

   (Incidentally there are probably half a dozen sites with the full text of Edwin Abbott's classic. Just insert "flatland" in one of the search engines - such as Google - and you'll get other snazzier versions of Flatland than the two at the Gutenberg Project.)

6. Charles Hinton was influenced by the English mathematician William K. Clifford, a English mathematician who died in his 30s and who doubted that space in the universe was "homoloidal" (another way of saying he didn't think that space was "Euclidean"). Clifford was a contemporary of Georg Riemann and was very interested in Riemann's ideas.

   Hinton was the the mathematician in America most involved with the fourth spatial dimension at the turn of the nineteeenth into the twentieth century. He gave us the term (still used) for a four-dimensonal cube: tesseract. The four paragraphs of the Introduction are worth reading, particularly the conclusion:

   "...Hinton's final assessment that we can only regard four-dimensional space as possible if three-dimensional mechanics fails to explain known physical phenomenon still rings true today. However, today we are dealing with a five-dimensional space-time instead of a four-dimensional space with a separate dimension of time."

   If you persist to the later essays at my site, you will realize that there is an ever increasing inventory of anomalies in deep space that are difficult to explain using three-dimensional mechanics in the conventional models (and one of them is impossible to explain in Friedmann space, if the initial observations are correct). Perhaps Hinton's century old advice will turn out to very relevant, after all.

7. Sten Odenwald maintains an excellent series of essays (many of them reprints from published articles over the years), although you will see the site has moved. A 1984 article on the possibility of additional dimensions is accessible from the link on Big Bang Cosmology.

8. The Fourth Dimension is a fine web page maintained by Eric Saltsman. Nearly everything you need to know about hypercubes and hyperspheres is right here. The links to short essays on "Non-Euclidean Geometry" and "Einstein overview" are worth the effort. I believe Eric and I started our respective sites at about the same time, back in 1997. The material at both sites has expanded remarkably from our original foray into this field.

9. Rudy Rucker's Home Page, who is now a professor at San Jose State University. Back in the 1970s while teaching in upstate New York, he wrote Geometry, Relativity, and the Fourth Dimension which, since there are so few books published on the subject, is still referred to quite often. The biography mentions this book, in addition to several other books, including a novel, Spaceland, which describes what happens when a creature from the fourth dimension visits Joe Cube (who lives in just three dimensions). Spaceland came out in the summer of 2002.

C. Other Links

1. The Fourth Dimension and Non-Euclidean Geometry in Modern Art by Linda Dalrymple Henderson. I could write the copy, but Linda did a much better job than I could via an e-mail received on March 13, 2002:

   I wrote a book in 1983 on the centrality of interest in higher spatial dimensions for early 20th-century artists--from the Cubists and Marcel Duchamp to the Russian Suprematist Malevich and even Surrealists. The first chapter contains what is still perhaps the fullest history of the early interest in the idea in the 19th century. Michio Kaku, for example, drew extensively on it in the art-related chapter of his book. It has been wonderful in recent years to see the resurgence of interest in this idea.

   MIT Press will reprint my book next year, with a long new intro. essay in which I'm tracking the fate of the idea in popular culture, science, and art from 1950 to 2000. Obviously, the Web has made a huge difference in the diffusion of related information.

   Also, you might find this article in The Austin Chronicle (July 19, 1999) a good elaboration of Linda's interest in this topic and how the book evolved.

2. Tony Robbin's Homepage. If you take the book link to"fourfield: Computers, Art, & the 4th Dimension", you'll see that the Forward was written by Rudy Rucker and that the Introduction was contributed by Linda Henderson. The 1992 copyright coincides with the resurging interest in the general topic. You might want to check out the lecture link too.
3. Einstein, Picasso: Space, Time, and the Beauty That Causes Havoc. This is a review of a book written the better part of four years ago by Arthur Miller which appeared in the December 2001 issue of Physics Today. One wouldn't think that Einstein and Picasso would be the subject of a scholarly book, and this shows the strange twists and turns non-conventional geometry has taken over the years in both art and science. The review gives the flavor of what was in "in the air" in the early part of the twentieth century.

1. Michio Kaku is generally on the "compactified" side of the ledger with respect to the fourth spatial dimension, and has written several books, the most interesting one in 1994 called "Hyperspace: A Scientific Odyssey Through the Tenth Dimension".
2. Nova: The Elegant Universe - Imagining Other Dimensions. The is the on-line site for the documentary on PBS early in 2004. In the section titled "How About 10-D?", there is a good explanation of how Oscar Klein envisioned the fourth spatial dimension as "curled up small". But another possibility is to assume that the fourth spatial dimension causes the lattice to lose its orthogonal relationship over larger areas. In other words, a "large" fourth spatial dimension might exist in a way that causes the lattice to appear as the surface of a sphere.

   In fact, no one right now knows beyond any reasonable doubt whether or not there is a fourth spatial dimension and, if it exists, whether it's "large" or "small". But while a "small" fourth spatial dimension has been investigated theoretically (via string theory), a universe with a "large" fourth spatial dimension has not yet received that much attention.

E. "Large" Fourth Spatial Dimension

1. Astronomy Bizarre: The Fourth Dimension is Back by J. Craig Wheeler, a teacher on the faculty of the University of Texas at Austin. The reference to "bizarre" - perhaps appropriate only a few years ago - seems less so today. This link was suggested by Linda Henderson (March 15, 2002), and she also mentioned:

   I loved hearing Craig Wheeler say to his students about embeddedness, "I used to tell students that asking what the universe might be curved into was not a legitimate question, but now we're interested in that topic."

2. Topological Lensing in Spherical Space by Evelise Gaussman, Roland Lehoucq, Jean-Pierre Luminet, Jean-Philippe Uzan, and Jeffrey Weeks. The "topological lensing" refers to the possibility of seeing the same object more than once - not necessarily in the opposite, or even the same, direction - and this unusual phenomenon is considered for virtually every known three-dimensional structure (some of them rather exotic). The Introduction in particular is worth a look; several basic concepts are defined clearly. I refer to this article a few times in the essays.

   [There's a downside, however, if you download the pdf version. It's 32 pages long (or 732 kb), and will take a minute or two to download. If you have an older computer with a relatively slow transfer rate, just about the time you conclude your computer had died and gone to heaven (or the other place), the tract will be displayed. There's a good illustration of the hypersphere as two spheres abstractly glued together on page 4 of the pdf version which, quite frankly, is a better depiction than what I did in Essay #1.]

3. Ring Around the Singularity is a synopsis of a more extensive article authored by R. Emparan and H. S. Reall in the fall of 2001 (with revision in the spring of 2002) and is available here. Notice the reference to a "large" fourth spatial dimension.

4. Thomas Banchoff's Home Page, who is a professor at Brown University (RI). If there is one person who can lay claim to being the world's expert on the fourth spatial dimension, he's it, though he seems to have concentrated on the "hypercube", rather than the "hypersphere".

5. Surfing Through Hyperspace by Clifford A. Pickover. This site gives the preface of his book published in 1999 and it's an excellent overview of the topic. Dr. Pickover writes: "Although the concept of the fourth spatial dimension is more than a century old, its strange consequences are still not widely known". (This is same sentiment I expressed near the end of the Introduction.) I certainly admire how Dr. Pickover frames the subject and then tells you how he is going to discuss it.
6. A Conversation with Physicist Brian Greene is written by John Fudjack in September, 1999, and has a interesting discussion of the alternate character of the fourth spatial dimension. (In the mid 90s Columbia physics professor Brian Greene wrote The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory, a surprising bestseller.)   In fact, there are a few quotes from my "old" series of essays in Fudjack's file.   By my biased reckoning, the most interesting part of the file begins when you scroll about one fourth of the way down, beginning in Section Three.

   [When John Fudjack was doing the research when he wrote this article, he knew that Brian Greene considered the fourth spatial dimension to be "small", as in string theory. Fudjack started using the search engines on the internet to look for sites which considered that a fourth spatial might be "large", and came across my site. While I can't say I'm all that interested in the psychoanalytic consciousness and the ilk - I'm a numbers guy! - the discussion is well rounded and quite fair with respect to the possibilities of that fourth spatial dimension.

   Incidentally, there are two links to my site at this site. The first link is OK, although the quote from my Introduction has been revised slightly in later editions. The second link at the end of Fudjack's file is to my old site in another city, which is long gone by now.]

F. "Medium" Fourth Spatial Dimension (Gravity)

1. Searching for Extra Dimensions is a short essay written by Greg Landsberg at Brown University. This nifty essay gives the rationale why physicists are searching for gravitational anomalies assuming that a "large" spatial dimension is probably about one millimeter (obviously not the same context as a "large" spatial dimension in Riemann's hypersphere, something that I mentioned in the Introduction).

2. This link on "Cosmological Constraints on Brane-World Cosmology" was mentioned in the introductory essay. The possibility that what we observe is "embedded" in higher-dimensional space has waxed and waned over the decades, but seems to be on the rise in recent years. What I am examining in these essays is "embeddedness" as it relates to Riemannian space (although this is not the only way "embeddedness" could occur).

3. An Invisible Dimension is a summary in Physical Review Focus of an article written by Lisa Randall and Raman Sundrum proposing that the fourth spatial dimension might not necessarily be "compactified". Perhaps there is a way of looking for it, as Lisa and Raman suggest.

4. Extra Dimensions is a short essay by John Terning. He presents the history of changing assumptions about the "extra" dimension(s). Once upon a time back in the 1920s, that fourth spatial dimension was assumed to be "small", as in string theory. Within the last five years, there has been a growing suspicion that it might not be quite so small, and that only some of the gravity that we observe occurs in a 3D "brane", that is, on the "surface" of a higher dimensional structure. The evolution of "extra" dimensions has reached the point where there is speculation that a fourth spatial dimension might be there in the same way it was introduced by Riemann in his doctoral thesis back in 1851.

   There are many valuable links at the end of John Terning's file which will keep you busy for awhile. They're not all relevant to what I'm concentrating on here, but one that is relevant is "4D Gravity on a Brane in 5D Minkowski Space by Gia Dvali, Gregory Gabadadze, and Massimo Porrati" where the authors state:

   "...we suggest a class of models in which 4D Newtonian gravity can emerge on a brane in 5D flat space. A crucial feature of these models is that the extra dimension is neither compact nor warped and its size is truly infinite."

   What I do in my series of essay at my site is provide an argument in topology which gives us that "infinite" fourth spatial dimension.
5. New Dimensions in Theoretical Physics is an article in Science Watch about novel concepts by Savas Dimopoulos and his colleagues. It's one of the better articles explaining why "extra dimensions" are necessary to explain gravitational anomalies in physics, and puts the previous Randall/Sundrum in better perspective. There are two linked files, and when you scroll about halfway down the first file, you'll see a paragraph referring to the Kaluza/Klein idea of how one more spatial dimension - the fourth one - could unify gravity and electromagnetism. Although tradition holds that a fourth spatial dimension exists in the way it is described in string theory, it's possible to derive a realistic cosmological model of the universe in which it exists as a "large" dimension, which is what I do here.

6. The search for extra dimensions was written by Steven Abel and John March-Russell in the November 2000 issue of Physics World. In the first ten paragraphs or so, it explains why a fourth spatial dimension allows Maxwell's equations governing electromagnetic radiation to be a specific solution of Einstein's theory of relativity. The fourth paragraph mentions a "rotation" presumably around some axis; Einstein and Kaluza thought it was the axis of a cylinder. It really has been an unanswered question for decades: in (or on) what type of structure does this "rotation" occur and what is its "size"?

   Near the end of the article it mentions the "extra" dimension in connection with gravity and mentions the upper limit for the size of this dimenson - again, about one millimeter.

As I recall, I had about five or six references in the "Links" file when I first published the essays over seven years ago. The list continues to grow, both in quality and quantity.