In ancient astrology, much was made of the Aspect, or angular distance between one sign and another. This ability to "see" or regard something was called an aspect. We still hear the term used to describe how someone looks in novels written in the 18th and even the 19th centuries.
The concept was based on the theories of Pythagoras, and involved what we now call Sacred Geometry, but also included their theories of optics.
These ideas were formulated more clearly by the Neo-Platonists and other philosophers, but the basic idea is that regular polygons were considered to be "pure" forms; they were even divisions of the circle that are calculated by whole numbers.
Light and therefore information could flow through these pure forms, but without being in the "line of sight", the object one regarded could not be seen. To understanding the basic meaning of a sign's "Aspect", just remember the "line of sight" concept and think of it in terms of "the ability to be seen".
The Elemental Trigons constitute one of these pure forms; signs of the same Element can "see" each other. The Quadratures, or Qualities of the signs can be drawn by marking out squares within the wheel of the Zodiac, and therefore signs of the same Quality can see one another. The Zodiacal Wheel can also be evenly divided by Hexagons, which show that all signs of the same Polarity can see one another. The last measurement was to divide the Wheel in half, meaning that signs placed opposite one another can see the sign at the opposite end of the diameter. Of course, each sign can see itself. The twelvefold division is the Zodiac itself - any larger numbers break the whole number rule.
It might seem that this covers all the bases, and that all signs have a way to see every other sign, but this is not the case. If we look at Aries as our first example, we already know that it can see Leo and Sagittarius, because they are members of the same Trigon. We know that Aries can see the members of its Quality, who are Cancer, Libra and Capricorn. From the hexagonal division, since Aries is an odd numbered sign (it is sign #1) we know it can see all other odd numbered signs. Libra is opposite Aries, and part of its quality as well. So who is left out?
Taurus is the second sign of the Zodiac. It is even numbered, not part of Aries' Trigon, and not a Cardinal Sign. Therefore, Aries cannot see Taurus. Gemini, is odd numbered, being the third sign, so it can be seen. Cancer is part of the Cardinal Sign group, and Leo is the next Fire Sign, so they can be seen. Virgo is an even-numbered Mutable Sign; it cannot be seen. Libra is a Cardinal odd numbered sign, and can be seen. Scorpio is a Fixed even numbered sign and therefore cannot be seen. Sagittarius is of the same Trigon, Capricorn is of the same Quality, and Aquarius is sign #11, an odd numbered sign, so the only sign left is Pisces, an even numbered Mutable Sign, and cannot be seen.
To explain these rules in simple terms, the signs adjacent to a given sign and its opposite do not aspect, or connect, to that given sign. Let's see what signs are out of aspect with each other.
Signs that have no connection by aspect usually have the most difficulty when it comes to getting along with one another. It's like one is constantly in the mental blind spot of the other. Sometimes the Moon and Planets in the birth charts of two people who don't "see" one another moderate this difficulty, but it's a good rule of thumb to ignore the idea that Elemental compatibility is a necessity, because, for instance, Aries and Capricorn are both Cardinal signs; they do aspect, or see one another.
This concept is also quite important when examining the planetary relationships between two people, especially in the case of the Moon. An example would be that if one person's Moon could see the other's Sun then there is a good chance that their differences would be easy to work out, despite the Sun Signs being out of aspect with one another. The aspects of the signs give us an easy way to define possible relationship problems quickly and figure out the best strategy for resolving them.