At first I was surprised to hear that Michael Schiavo was requesting an autopsy on his wife’s brain; now I think I’m beginning to understand.

George Felos, the attorney for Schiavo’s husband, Michael, has said that a Florida medical examiner has agreed to perform an autopsy after Terri Schiavo dies. Florida …

The box I like to put God in won’t contain Him….

Easter came early this year, and with it another church musical program. (I hesitate for many reasons to call them “cantatas” ;o) I rushed. I worried. I made programs (too many) and harrassed all of our participating members with daily (sometimes hourly) e-mails in an attempt to encourage people to …

A couple of weeks ago I was discussing spiritual gifts and ministry burdens with a dear friend and sister in Christ here at the IBC in Boston. She posed a question that no one has ever really asked me before: “What are your spiritual gifts, Jen? What is your bent …

It’s pretty hard to dismiss a kid’s enthusiasm. Even when a five year old rings my doorbell three times right after I’ve just woken up Saturday morning (pre-coffee, mind you) and screams “HI Heidi!!! Look at my new toys and…

jonquils crane their necks to peck pick preen another’s face ignore the sun….

So this morning I had to give a devotional to the high school teachers here at the school where I work. I came home from church last night tired and exhausted. After a little bit of downtime (think CSI:NY), I gathered some things and began trying to decide what I should speak on today. I started by going through several books of ideas that usually get me going into some line of thought appropriate for a devotional, but nothing was coming. I tried a few more books… but nothing. Eventually I just sat there looking at my bookshelf, staring at the books, and hoping for a source of “inspiration.”

But nothing came.

Then, my eyes happened to fall upon a new book that I just picked up this last Sunday - True Worship by Ward and Whitcomb. Now, I’ve barely even opened the cover to the book, but it got me thinking. Worship happens to be a topic that has interested me in the last year or so, and with the adjective “true” in front of it, it started me thinking. What is true worship? As I began thinking through Scripture, my mind landed on the idea for my worship devotional - personal encounters with God and their results. I present a brief synopsis of it here for your edification:

Exodus 3. The burning bush incident. Moses is instructed to remove his sandals because he is standing on holy ground.
Exodus 19.16-25. God appears to the nation of Israel at Mount Sinai. Moses is instructed to establish a boundary around the mount so that none of the people get too close to God’s holiness.
Exodus 33.18-23 and 34.29-35. Moses sees the back of God’s glory. Moses’ face requires a veil whenever he is in the presence of other people because his face is shining too brightly.
Isaiah 6. Isaiah sees the Lord “high and lifted up.” He is immediately struck by his own sin and his nation’s unworthiness.
Matthew 17. Peter, James, and John see the Christ transfigured before them. Peter’s initial response (however incorrect it might be) is to build a place of worship there on the mountain.
Revelation 1.7. Christ will return someday. Every one on earth will see Him, and all tribes of earth will wail (mourn).
Revelation 1.12-17. John sees one like a Son of Man standing in the middle of the seven golden lampstands. John falls at His feet as if dead.

Now, this list is by no means exhaustive. There may be other passages that should have been included in this listing, but due to lack of time I’ve not yet spent time looking for them. (Feel free to add them as a comment, if you want!) However, in every one of these instances, seeing God for Who He is makes a noticeable and normally physical change in the life of the one viewing Him.

The thought for today then is this: How often do you truly see God for Who He is? We talk so much about “worship” today, but do we really have true worship? Do we see the God of the Bible or a god of our reasoning and imagination? When was the last time you were physically affected by worshipping God?

What about those “worship services”? When was the last time you walked out of church changed because of Who God is? If God isn’t worshipped, please don’t blame the music director… the musicians… the preacher… the other members… or even the congregation at large. If God isn’t worshipped, it’s usually because you and I weren’t looking for Him. We don’t want to worship Him… because true worship will change us.

So… when was the last time you truly worshipped?

For some reason the thought struck me today that a spatial graph does not need to be confined to three dimensions. In other words, the fourth dimension and beyond need not be some mysterious and barely comprehensible phenomena. Each variable in an equation can be interpreted in a variety of ways, and spatial coordinates happens to be one of the more well-known of those ways. But traditionally the fourth dimension has been interpreted graphically as some sort of weirdness resulting in such items as the enigmatic “hypersphere” and others.

The first thought that I had on this subject was simply that our standard graphing coordinates rely on the position of axes such that the same scalar value on each axis (a single point) is equidistant from the others, i.e. x = y = z = K where K is any number. Connecting these points results in the plane x + y + z = K. My feeling was that by simply adding another axis and repositioning the four of them so that w = x = y = z = K for any K, one could then graph any given object in four dimensions. Once this is proven, one could likely demonstrate that any number of axes could be arranged so that any object could be graphed therein. This would simply be another interpretation of multiple dimensions—one where the traditional mystique of the fourth and fifth dimensions, etc., is discarded in favor of a system whereby any number of dimensions could be represented spatially. I have made several observations already about the possibilities of this system, but I can already see that it is fraught with difficulties that only mathematicians far more accomplished than I could fully confront. In fact, I believe someone has very likely thought of this theory before me, but I am not aware of that being the case.

One observation is that this system, particularly the arrangement of axes, can be constructed from the primary “nonexistent” 2D elements: points, lines, and planes. One end result of this is that the 0, first, and second dimensions remain theoretical and non-spatial, while every dimension from the third onward is spatial. I wondered for a while how I could determine the position of the axes for the fourth dimension, then I finally had a thought. The 3D coordinate system has three axes, which can be considered as six “sticks” protruding from the origin. You can create symmetry between three of these sticks and the other three by separating them with a plane. If you were to view one side of this plane from the origin point, you would see three lines extending outward, each of which is equidistant from the next and previous line. I wish I could provide a picture, but I am a bit too lazy to draw one up right now—however, I plan to do so in the future. However if you can see these lines extending from the origin point as I am describing, you will notice that connecting the tips of the lines as they extend out results in an equilateral triangle. And this is precisely the graph resulting from x + y + z = K where the value of K is the same on each of the three axes and the graph is limited to one of the eight octants of the coordinate system. So now what would happen if we added a fourth axis? My thought is that if we divided the resulting eight “sticks” with a plane, as we did with three axes, and positioned ourselves at the origin, we would see four lines extending, each of which is equidistant from the previous and next. So in this case a square is projected.

The major problem that arises from this is that for any value K, the value of K on each axis is not equidistant from the other three, but only from the next and previous axis. I am not sure at this juncture whether the possibility exists for arranging four points such that each one is equidistant from all the rest. My thinking is that it is not possible, and I am uncertain whether this would ultimately matter—I do not think it nullifies the plausibility of my system, but it will result in variant graphs for equations that may have the same form. In other words, z = y + x may look different from z = w + x even though the forms are the same. This does not come as a major surprise, since the 3D coordinate system is arranged such that the angle between each “stick” and its neighbors is 90 degrees. This right angle property of the 3D coordinate plane facilitates a level of simplicity and intuitiveness that additional dimensions would lack in my system, but only for the majority of the basic shapes to which we are accustomed. Certain crystalline structures may be much simpler to graph in a supratridimensional coordinate system.

Inconsistency from one dimension to the next is expected. For instance, the graph x² + y² = K in two dimensions results in a circle, but in three dimensions it is a cylinder. A circle in three dimensions is an intersection of two graphs: for example x² + y² = K and z = 0. Similarly a 3D sphere x² + y² + z² = K would look much different in the 4D coordinate system. My initial thought was that a typical sphere could be graphed in four dimensions as w² + x² + y² + z² = K, but the aforementioned lack of equidistance between each axis from the other would preclude this. However my only goal is demonstrating that any number of dimensions can produce purely spatial graphs.

Returning to the projected shapes I mentioned earlier, you can see how this can easily extend to more dimensions. Five dimensions would result in a projected pentagon, six dimensions in a hexagon, and so on. The angle between each of these axes and the dividing plane would seemingly be equal, but the angle of each stick in relation to the others would differ by varying amounts (to be determined later by me or someone else perhaps). This projected shape must appear on any given arrangement of sticks that are all on one side of the dividing plane. As an aside, I made sure this theory was “backwards compatible” with two dimensions, which it is: in this case a line is projected, but it is separate from the lines that comprise the axes. In one dimension a point is projected.

Another pattern I noticed is the number of projected figures that can appear on the coordinate systems. A 1D graph has two sides, a 2D graph has four quadrants, and a 3D graph has eight octants. The computer programmer in me immediately notices a binary squares pattern. 1D = 2¹ projections, 2D = 2² projections, 3D = 2³ projections. I am hypothesizing that a 4D coordinate system must contain 2⁴ projections, or 16 squares. (Coincidentally I have for a long time been fascinated with the number 16 and its frequent appearance in nature and society. This is merely another example of its ubiquity.) Now looking at a 2D graph of its projected lines (|x| + |y| = K) from its normal perspective results in a tilted square (or a 2D diamond). Similarly a 3D graph of all eight of its projected triangles pieced together results in an eight-sided diamond. So I might expect that the 4D shape formed from its projections would be a 16-sided figure where each side is a square—or perhaps that would be the case only if the aforementioned “equidistance problem” did not exist. As of now, I can only envision a typical cube as the result of connecting the dots of four intersecting axes, which is a six-sided figure, but I feel like I am missing something. Are the rules different for even-numbered dimensions? Or should I be trying to envision “2D diamonds” rather than “2D squares” even though they are essentially the same? Or is the occurrence of triangles in this figure impossible to escape? Right now I am not sure about these questions. I only know that my basic concept of a dividing plane and projected 2D shapes (where the number of sides is equal to the number of dimensions) seems to make sense and may be a potentially useful method for constructing supratridimensional coordinate systems that can render entirely spatial graphs.