The Tao begot one,
One begot two.
Two begot three.
And the three begot the ten thousand things.

The ten thousand things carry yin and embrace yang.
They achieve harmony by combining these forces. (TT Ching, Ch. 42)

How did Lao Tzu arrive at the at number 3 as the underlying principle of order?

Most likely by observing a certain threefold manifestation in nature. Examples: (heaven, earth and water); (birth, life, death);(past, present, future); (thought, word, deed);( man, woman, child); (three sides of triangle); (primary colors; yellow, red, blue)(water: liquid, solid, vapor) and many others. Lao Tzu himself says:

Look, it cannot be seen--it is beyond form.
Listen, it cannot be heard--it is beyond sound.
Grasp, it cannot be held--it is intangible.
These three are indefinable;
Therefore they are joined in one. (TT Ching, Ch. 14) (See also: TT Ching, Ch.19 & 67)

From the above we can see that Lao Tzu sees sight, hearing and touching as means of expressing the presence of the non-sensible Way. Andrew Harris has well said: All things that are especially complete and entire are stamped with the number three. (Website: Symbolic Meaning of Three) But Lao Tzu's statement "Three begot the ten thousand things" seems to postulate an order of three before creatures can multiply and diversify.

Hints to the Triune nature of God are found in Genesis where we see God creating by his word and the mighty wind sweeping over the waters and where we read, "Let us make man in our image" (Gn 1:26)

In Christian teaching, the Father Creates through the Son in the Holy Spirit, but only recently have we begun to probe its deep significance. "From the continual inflow of the divine Spirit (ruach) created things are formed (bara'). They exist in the Spirit, and they are 'renewed' (hadash) through the Spirit. This presupposes that God always creates through and in the power of the Spirit, and that the presence of the Spirit therefore conditions the potentiality and realities of Creation." 64

If the Spirit is the vivifying power of the universe He does it according to the pattern of the Word who is continually creating with the Father because "He is ...the firstborn of all creation." (Col. 1:15) Man made in God's image is more precisely made in the image of God's Son in his humanity because destined to become a son/daughter of the Father. The Word is the pattern of law, order and grace because the world is being created with him, in him and for him. Teilhard de Chardin has well written, "Christ invests himself organically with the very majesty of his creation" 65 and becomes its very point of attraction. Now the faith, hope and love of human beings, or shall I say, all intelligent forms of life in the cosmos are destined to gravitate toward the Son God God.

Action of Tao