Equal-Area and Equal-Shaped Surface Sectors on Spheres

During times when I’ve been engaged in mundane, repetitive chores or assignments, there’s been a tendency for me to speculate on things that don’t necessarily connect directly with my life, but still hold some interest for me. For example, I’ve allowed myself to be nagged with an apparently trivial problem that someone out there (maybe one of those math “educators”) has probably trained himself or herself well to be able to solve. And that problem is the following:

If you take any globe and divide it by lines of latitude and longitude, then you will notice four things:

--Each line of latitude intersects each line of longitude at right angles.

--All latitude lines run parallel to each other.

--All longitude lines converge at the North and South Poles.

--The areas bounded by these lines (separated an by equal number of degrees) are greater the closer one gets to the Equator and smaller the closer one gets to the Poles.

Now my question is this: How can I devise a way of dividing a globe (or sphere) into sectors (of the surface) that are shaped in the same way and have the same area? Other then completely disregarding latitude and dividing sectors by lines of longitude, I can see no apparent solution to this. I’ve read about geodesic spheres, but the projections on their surfaces are not all equal to each other, only approximate at best. Is there an intuitive answer that would have been obvious had I not been trained to be oriented toward straight lines and two-dimensional grids? Could this be just one example of shortsightedness when it comes to training children in spatial perception and reasoning? Well, I’m throwing out the problem to some of you hard-working homework-doing alumni/scholars out there! Comments, either in the form of solutions or perplexed reactions, are welcome!

And there are a couple of other questions that I’d like to toss out while I’m at it. What, mathematically, is the difference between the celestial “sphere” as I look around me in any direction (counting “looking” down at the Earth as part of it) and the surface of a three-dimensional sphere? And furthermore, is there really such a thing as an absolute shape (since it appears that any shape depends upon the perspective of the observer)?

There is actually rhyme and reason behind these questions. There is a certain disconnect I see between the real world as I perceive it and the way people have been trained under formal education to quantify and analyze it. First we learn to count numbers and model our reality as if it were a one-dimensional number line. Then, the more advanced we go into math education, the more we learn to interpret reality as a two-dimensional plane. This would be suitable if we all lived in Flatland! Even our so-called computer-generated three-dimensional models are only projections onto a two-dimensional screen. We seem stuck in a two-dimensional framework with straight lines and an “objective” frame of reference that denies the perspective of the observer. It is the transcending of these limiters to thinking that interest me a great deal. I want to think with a subjective three-dimensional understanding on a complex level. In other words, nothing less than a complete overall of how I view things!