Here are some new go proverbs from Japan. They are due to Hidaka Toshiyuki 8-dan, one of the pupils of Kitani Minoru.

Renraku wa ni ga kihon.
Two is the magic figure for connections [in fighting].

This refers to Manhattan distances of connected stones: kosumi and ikken tobi are connected by a count of 2, keima and niken tobi by a count of 3. Twos are better for contact fighting. The idea is to play your stones in such a way that you create kosumi and ikken tobi shapes for yourself and keima for the opponent.

Manhattan distances are where you go orthogonally along the marked lines, as in walking round New York streets, instead of taking diagonal short cuts.

After White attaches and crosscuts, his stones are in a keima relation (bad) to his stones on the outside. Starting at the other two points and following up with a similar crosscut would lead Black into a difficult fight where he would need other friendly stones nearby to help him out. Here Black has the enviable choice of playing on the other side with a, b, c on move 5.

Using the proverb you can tell instantly where of a, b, c or d to invade for Black: a or c is correct.

Jouseki no kihon wa daidaigeima.
The ideal basis for a joseki is daidaigeima.

Groups that are (partly) excluded from the corner have to develop along the side. For them the daidaigeima is often the ideal extension. See many josekis. Black 7 below is a daidaigeima (very large knight's move) from 1.

Hiraki no kihon wa goken.
The ideal opening extension is 5 spaces [between stones].

This applies both to formations such as extending from a shimari (corner enclosure) or to the sanrensei (three star points) formation.

Ishi hitotsu ni taishite ikken hanasu.
[When attacking with a pincer] leave a one stone gap for each stone [in the enemy wall].

This refers to the middle game, not joseki, and is a modern corollary to the ancient proverb that tells us to extend n+1 points from a wall of n friendly stones.

Example 1: Because White has three stones the correct place for Black to attack is three spaces away at 1.

Example 2: Black should attack at 1 or a. This is because White can jump to 2 which is effectively sente, forcing 3. White's wall is then two stones, so Black's pincer should be two spaces away. And obviously against a single stone you should be one space away.