You can use your mathematical prowess to add metamagic effects to your spells without using a higher-level spell slot.
Benefit(s): When you take this feat, select two metamagic feats you do not yet have. When casting a spell, you can perform the steps below to spontaneously apply the effects of either or both of these metamagic feats, as well as any other metamagic feats you have, to the spell without expending a higher-level spell slot. Using a feat in this way increases the spell's casting time to the casting time it would take if the character were a sorcerer or bard (sorcerers and bards using this ability increase the spell's casting time by two categories); applying the Quicken Spell metamagic effect negates the increased casting time but still contributes to the modified spell's effective spell level. You can take this feat more than once; each time, select two additional metamagic feats, adding their effects to the list of possible effects you can apply to spells with this ability.
When casting a spell using Sacred Geometry, first determine the effective spell level of the modified spell you are attempting to cast (calculated as normal for a spell modified by metamagic feats). You can apply any number of metamagic effects to a single spell, provided you are able to cast spells of the modified spell's effective spell level.
Refer to the Prime Constants table to determine the prime constants that can be used to cast a spell of the desired effective spell level. Then roll a number of d6s equal to the number of ranks you possess in Knowledge (engineering). Perform some combination of addition, subtraction, multiplication, and division upon the numbers rolled that gives rise to one of the relevant prime constants. If you can produce one of the relevant prime constants, the spell takes effect with the declared metamagic effects, and you expend a spell slot of the unaltered spell's level. If you are unsuccessful, you fail to cast the spell, the action used to cast the spell is lost, and the spell slot is used up. The DC of any concentration check to cast a spell affected by this feat uses the effective spell level used to determine the prime constants, even though a successful casting of the spell does not expend a higher-level spell slot.
For example, suppose a wizard wants to add the enlarged and empowered effects to the magic missile he just cast. He adds the total level increases of the metamagic effects to the spell level: 2 [empower] + 1 [enlarge] + 1 [magic missile's base level] = 4, so the spell's effective level is 4. The wizard refers to the Prime Constants table, and determines that the prime constants for that effective spell level are 31, 37, and 41.
He then performs the following operations to give rise to one of the prime constants he needs: (6 × 6) + (4 – 3) × 1 = 37.
Note that he could also combine the numbers as follows: ([6 + 6] × 3) + 4 + 1 = 41. He has produced one of the prime constants for the desired effective spell level, so his empowered enlarged magic missile takes effect after the full-round casting time, and uses only a 1st-level spell slot.
If the result of his dice pool were 1, 1, 2, and 5, he would have been unable to produce any of the relevant prime constants. His attempt would have been unsuccessful, and he would have expended a full-round action and his prepared 1st-level magic missile spell with no effect.