We trinitarians tend to be over-reliant on analogy. Now analogy is important because we are discussing a topic which is unperceivable. However, we tend to present the analogy saying "Trinity is" instead of "Trinity is like" therefore denoting equivalence instead of similitude. So, here is a list of several analogies of Trinity and how they break down:
- Apple:An apple is skin, core, and fruit, but one apple.
The problem is that there is a partition here. In the Trinity, the Father, the Son, and the Spirit are distinct, but not separate. In other words, there is a sense that the Father is other than the Son, but there is no point where the Father stops and the Son begins, they are united. Furthermore, there is the issue of variance. The Father, Son and Spirit are one being, but also one nature, in that they are all perfect, and do not different in attribute. Skin, core, and fruit are far too different.
- Stream:You can have a single river that, at a point, spits into three streams, that then come back together (like two parallel islands are in the river). At this point, it is still one river, but exists as three.
This avoids the variance issue above, however, there is still the problem of partition. Additionally, if presented improperly, it can be used to support a change in God.
- Water: H2O can exist in the same environment as ice, liquid, and water vapor. However, they all remain water.
If "in the same environment" isn't mentioned, this alludes to Modalism, and still can be viewed that we if presented improperly. Additionally you have the variance issue.
- Synonyms: Example: Dual, Double, and Binary are different words, in that they have distinct uses, however they all have the same meaning.
This both overemphasises the unity and the distinction depending upon presentation. In the overemphasis on unity, there is a sense that the Father, Son, and Spirit are only a titular distinction, and not a substantial one. The distinct use point is presented to counteract this because there are times when the use of double is appropriate and the other two are not, and likewise for the other two.
The overemphasis on distinction is what is known as alienation. This means you can have one without having the other two. Again, if presented correctly, this is avoided by saying that if something is only appropriately described as binary, there still exists a doubleness and a dualness. But this aspect is still quite weak.
PS, please refrain from commenting on the numerical nature of the example. It was only used because of the lack of connotational difference.
- Jar o' Gas:Take three gases (for instance Oxygen, Nitrogen, and Neon) and put them into one jar. Though they are three gases, they are completely unified.
there of course is still the variance issue (Oxygen and Neon are very different). However, this also introduced what is official called "confusion", but better understood as blending or mixing. this implies such a strong unification, that it becomes impossible to differentiate one from the other. Variance is still the stronger issue, for if you get down to the molecular level, confusion ends. But there is confusion on the Macrocosm.
If you look at it at the microscopic level to avoid the confusion issue, you end up with the alienation issue, for Oxygen knows nothing of Neon, etc...
- Lamp: There is a lamp with three lightbulbs. When turned on it produces a single light. (This analogy came from Thomas Oden in his book The Living God.)
We have two sets of three here, and there are different problems pertaining to what you emphasize. If you focus on three lightbulbs, one lamp (which is incorrect considering the phrasing), you have the partition issue.
If you focus on the light (one light, three sources) you have the issue of confusion, where you cannot distinguish which photon came from where. If you change it so that each bulb is a different color (ignoring the variance issue this creates) its still doesn't end the confusion, for they merge as one color in the light.
A more, overarching problem is a reversal of revelation: you have three revealed as one, instead of one revealed as three.
- Humans:Humans are mind/body/spirit, yet one being
This one is a favorite primarily because unlike everything else mentioned, it is actually a trinity. You have three distinct substances (hypostasis) yet one essence (being). However, this is still an analogy towards the divine Trininy because, (a) we are not God, and (b) though we are a substantial trinity, we are not a trinity of persons. A person is a substance with a rational nature, and the body does not have this. Mind and spirit divide the rational nature, one having the ability to think, the other containing will. With mind and spirit you have soul, which is your "personhood" so to speak. (I'm not really an expert on anthropology so i remain correctable on these finer points)
One other difference from the divine Trinity is, once again, variance. Now variance doesn't undo the one being/essence aspect which is necessary for trinity. However, it does contradict the one nature aspect which is necessary for perfection in the divinity, and besides its just incorrect.
Some Trinitarians have been know to overemphasize this analogy (I was guilty of this myself once) by saying that the Father is the mind, Son the body, and Spirit in spirit. This is given credence by the "mind in his image" aspect of humanity. However this limits the personal nature toward each member of the Trinity and is inaccurate.
The point of these analogies are to present elements of the concept of the Trinity, not to create the whole idea for you. A three dimensional example is an impossibility (human except due to our spiritual nature). Therefore, in order to properly understand it, it does take effort on the part of the listener to step beyond the analogy to that which is trying to be expressed.