Tier: First
Computation: 1 segment
Area of Effect: 1 pane per level
Range: 3// per two levels
Duration: One turn per level
Glass Required: Yes
B-M-S Cost: 1.0
This function creates a perfectly flat and vertical two-dimensional plane of geometric force. It is 95% transparent, and can be tinged with any hue the Mathematician pleases. The plane measures 1// by 1//. The Mathematician can create one such plane per level of experience and each plane has one structural point. The plane is proof versus all physical attacks (until broken down by sustaining damage), and the armour class of the segments is AC(5) with a +1 bonus to AC for every third level above level three. The shield can be initially placed anywhere within the casting range, but afterwards can be moved at 1/2 the Mathematician's normal ground movement. The "movement" is relative to the Mathematician's position, thus, if the Mathematician is standing on the deck of a sailing ship that is traveling at 9// per round, the shield panes will not drift away behind the boat.
Tier: Second
Computation: 3 segments
Area of Effect: Sphere: ¼// diameter per level
Range: 1// per level
Duration: One round per level
Glass Required: Yes
B-M-S Cost: 1.25
The Geometric Sphere is used primarily as a defensive combat function. Often it is used to capture enemies and contain them. The Sphere has a ¼// diameter per level. The sphere is gas permeable, and so does not cause asphyxiation. Moreover, it causes no damage to partially enclosed creatures (i.e. a 6' tall man enclosed by a 5' diameter sphere would undoubtedly be immobilized, but would otherwise take no damage.) An unwilling target gets a save versus area effect to avoid being enclosed. Large spheres can enclose more than one person or object. The sphere can be moved at the Mathematician's will at a rate equal to 1/2 the caster's base ground movement. Like most "Geometric" formulae, the Geometric Sphere has armour class of AC(5) with a +1 bonus to AC for every third level above level three, and has a single structural point.
Tier: Second
Computation: 2 segments
Area of Effect: 1 pane per level
Range: 1// per level
Duration: Two turns per level
Glass Required: Yes
B-M-S Cost: 1.0
This function is quite handy in areas that have poor footing or are heavily populated by traps. Like the Geometric Shield, the walkway is composed of 1// square transparent panes. Each pane can withstand 150 lb. plus 15 lb. per level of the Mathematician. The panes have but one structural point each, and is AC(5) with a +1 bonus to AC for every third level above three. The panes are always horizontally oriented. Each pane can be arranged in any manner the Mathematician, thus the walkway can take 90° turns in its path. If more than one pane is summoned, then each pane must touch; diagonal tangents are permitted. A mapping function is often useful when calling upon this power. A Walkway must always be flat. If a vertical component is needed, then the Mathematician must implement a Geometric Staircase.
Tier: Third
Computation: 5 segments
Area of Effect: Cylindrical volume, see below
Range: 1// per level
Duration: 2 rounds per level
Glass Required: Yes
B-M-S Cost: 0.75
This formula brings into existence a tube whole length is 1// per level of their Mathematician, and whose diameter is 5' plus a maximum 2.5' per three levels. The tube is segmented into 1// lengths and has one structural point per three levels of the mathematician per 1// segment. The tube shell has an armour class of 5 and is impermeable to gas, liquid or solid unless its structural points are overcome. The tube can be summoned into being with both ends open, both ends closed, or one open and one closed. For very long tubes, the Mathematician can command the tube to turn at precise 90° angles at 1// segment lengths.
This formula cannot be used to breach another Mathematician's Geometric Shield/Sphere/Walkway, etc. Thus, it cannot become an "instant portal" through a hastily erected defense. However, if the caster wishes to place a tube portal through a shield of his own creation, then that is permissible.
Tier: Fourth
Computation: 1 round
Area of Effect: See below
Range: 2// per level
Duration: One turn per level
Glass Required: Yes
B-M-S Cost: 2.0
Great for siege operations, this function creates a temporary staircase so that high walls may be overcome. Each step in the staircase has a minimum dimension of ¼' high, ½' wide, and ¾' deep, while the maximum size of each step is 3' high, 5' wide, and 2' deep. The Mathematician can create two steps per level, and each step can withstand a maximum of 100 pounds plus 50 pounds per level. Each individual step has but one structural point. The destruction of individual steps does not collapse the staircase; instead, the steps above the one destroyed slide down to fill in the gap.