Amp Spire

=Gameplay Notes=
 * Unlocked at Rank 12
 * This tower increases the damage of other towers within its range by a percent amount.
 * If a tower is within the range of multiple AMP Spires, the tower will only be affected by the AMP Spire granting the highest increase.
 * The area of effect of the Amp Spire increases with its rank. For ranks 1 and 2, this area is a 3x3 square around the spire. For level 3 Range Spires, the area of effect is a 5x5 square without the corner blocks (another way of stating this is "two straight blocks away, or one diagonal and one straight block away, but not two diagonal blocks away")
 * Its range can be increased by the Range Spire.
 * This tower is paramount to players wishing to get far in Survival mode.
 * Overcharging this tower increases the damage increase effect by 1 percentage point per 50 resources invested.

=Optimal Usage Calculations=

Introduction
Distributing resources between your Damage Towers and the AMP Spire is always a balancing act. Due to the nature of how the tower's DPS synergizes with the AMP's percent damage increase there is no one fixed distribution of total resources between Damage Towers and the AMP Spire (i.e. always 80% to Damage Towers and 20% to the AMP Spire). In fact the optimal distribution keeps shifting the more resources are available for allocation. It is of note that the average DPS per resource of the built towers has no effect on the optimal distribution.

The following is a detailed discussion of the underlying calculation method. If you only wish to quickly calculate the optimal distribution, you can skip to the Reference Table further below.

In-depth Explanation
The following piecewise-defined function describes the relationship between the amount x invested into the AMP Spire of the total resources c and its effect on total DPS dealt by the towers (including the AMP Spire's bonus):



x: Amount of resources allocated to the AMP Spire

c: Total amount of resources available for allocation between Damage Towers and the AMP Spire

a: Average weighted base DPS per Resource of all Damage Towers impacted by the AMP Spire's bonus

Note that the parameter c only comprises the total amount of resources available for allocation between the AMP Spire and Damage Towers impacted by the Spire's range. Resources that go towards non-damaging towers (i.e. Slowfield Dispenser or Range Spire) or damaging towers not in range of the AMP Spire are not counted towards the parameter c.

The following shows an exemplary curve plotted using the afformentioned function:



In this example a is set to 1 and c is set to 6000. The red lines indicate the position of x at which the function's value is maximized. In other words, in the case of c=6000 a budget of 500 resources must be allocated towards the AMP Spire and the rest of 5500 resources must be spent on Damage Towers in order to maximize the total DPS. Note that an increase or decrease of parameter a will only affect the total DPS, but has no impact whatsoever on the optimal distribution of resources. This means in the case of c=6000 the optimal distribution will always remain 1:11 (500/5500).

Reference Table
''Parameter c: The amount of resources that will be distributed between Damage Towers and the AMP Spire. This only applies to Damage Towers within range of the AMP Spire and does not include non-damaging towers such as the Range Spire or the Slowfield Dispenser.''

''Parameter x: Indicates the portion of c that needs to be invested into the AMP Spire for optimal DPS. Naturally, the remainder (c-x) must be invested into the Damage Towers.''

Due to the dynamic relationship between parameters x and c together with the complexity of the function itself, calculating the optimal resource distribution on-the-fly becomes quite unwieldy. For convenience's sake, a reference table has been calculated using appropriate software. Simply choose the value for parameter c that approximately matches your current budget and you can read the optimal value for x.

1for values of c that exceed 1'000'000 --> x= c/2