Volumes of Tori and Pyramids

The volume of some solids can be found intuitively (e.g. for a right rectangular prism, V = lbh). However, some more exotic solids will require calculus for their volumes to be found.

Two such solids are the torus (donut) and rectangular pyramid. The former's volume can be found, possibly among others, using the technique of volume by rotation, and the latter's with volume by cross-section. These techniques will be used for deriving the formulae for the volumes of the respective solids. In summary:

• Volume of torus = 2cπ²r²
• Volume of rectangular pyramid = ¹/₃abh

WoW Mathematics: Sharing the (DPS) load

In any (conventional) boss encounter, a certain number of damage dealers (dps) need to whittle down the boss' health to zero over a certain time period. Barring any extraordinary tinkering by Blizzard, the health level remains constant between attempts.

This fixed DPS burden needs to be shared between all dps. I will examine the DPS burden per dps and how it changes with the number of dps that happen to die during the encounter.

Summary:

• Given a total DPS burden (total raid DPS) of D and a total number of dps of d, the average DPS burden per dps B, where x players are dead, is B = ^D/_{d - x}.
  

WoW Mathematics: Fatally Attracted to You

Back in the Burning Crusade era, Mother Shahraz was notorious for the encounter's high Shadow Resistance requirement and her femininity. She would also, at random times, cast Fatal Attraction on three random non-tank players.

Despite having Shadow Resistance at the cap, the damage is still great, and the unlucky trio need to separate promptly if they are to survive. It may be useful to know, given a certain set of vectors and using calculus, how fast they are moving away from each other...

WoW Mathematics: My Large Brown Sack

To help find items easily, players often organise them in their bags in a certain way. It might be interesting to know just how many ways those items can be arranged, and what happens of those arrangements are restricted in some way.

Permutations are very useful in this process. It is a special case of the fundamental counting technique where an object, once picked, cannot be picked again, and so the available pool of objects decreases by one each time. Consider a certain 10-slot bag, the Large Brown Sack...

The Spectrum of Classes

This spectrum diagram illustrates the position of WoW's nine playable classes in the class balance. It is two-dimensional: rigour is read along the horizontal axis, while tactic is read along the vertical axis.

Rigour: Any given class uses a certain balance of physical abilities (using a weapon and/or shield) and magical abilities (casting spells) in combat. The classes to the right place an emphasis on the magical, with the priest and mage being magically pure. Classes closer to the centre are hybrids, who appreciate a combination of the magical and physical. The druid enjoys most a balance of the two rigours. The mana line divides the spectrum into those who use mana and those who use other power sources. To the left are the classes who place emphasis on the physical, with the warrior and rogue being physically pure and to the left of the mana line. Hybrid classes near the centre can tweak their respective balances through talent specialisation, though pure classes at the extremes are resolute about their respective masteries.

Tactic: Any given class uses a certain balance of tenacity and ferocity when engaging in combat. Tenacious classes will mitigate the damage taken in an attempt to outlast the opponent. The tenacious warrior will reduce the damage taken initially through heavy armour, while the tenacious priest will heal the damage that is taken. Ferocious classes maximise damage dealt to end combat as soon as possible. The rogue and mage place emphasis on this. Tactic can be tweaked by all classes through talent specialisation. The classes at the junctions (warlock and hunter) enjoy the versatility that class pets bring, so as to exercise a level of control over this balance.

It would be important to note that I cannot find a place for the new death knight class in this spectrum. This may be due to the youth in their class development; Blizzard simply has not yet found a niche that they can fill. It may suggest any of the following:

1. There are no more niches that a death knight (and future hero classes) can fill, and they are doomed to be patchworks of any of the nine original classes. In fact, death knights possess some of the qualities of paladins and warlocks.
2. The domain of the death knight and other hero classes lies outside of this spectrum.

Succeeding at left and right

The Thaddius encounter involves lots of movement, though its principle is very simple. Reviewing the guild's convention:

• Facing the boss from the entrance, positively charged people go to the right.
• Facing the boss from the entrance, negatively charged people go to the left.
• When changing sides, stick to the left side while you are approaching the other side. This will only be necessary for as long as people react slowly to Polarity Shifts.

My mnemonic for the first 2 points is the mathematical convention for number line graphing. The positive direction is right from the origin, while the negative direction is left from the origin. My mnemonic for the 3rd point is Australian road rules. Keep at the left side of the road!

The hardest part of Phase 2 is figuring out which side you are supposed to go to for the first Polarity Shift. Get ready to change sides if you need to. Facing the wall helps, as this helps orient you according to the left and right directions.

Polarity Shift is then cast. You have 4 possible consequences:
1. If you are at the left, and you are positively charged, you switch sides.
2. If you are at the left, and you are negatively charged, you stay.
3. If you are at the right, and you are positively charged, you stay.
4. If you are at the right, and you are negatively charged, you switch sides.

Provided you take the right course of action, it is smooth sailing until the boss down. Simply change sides when your polarity changes. No more worrying about positives and negatives!

Originally posted here.

Don't slack off during the Thaddius encounter!

Aim: To calculate the drop in DPS of a single death in Phase 2 of the Heroic Thaddius encounter, subject to certain assumptions. To observe the change in total raid DPS as additional dps die.

Summary:

• The minimum personal DPS needed to down Thaddius in time is 2186 DPS.
• The total personal DPS is 100% average DPS + 115% bonus DPS from charge buff OR 215% of average DPS.
• A 1-stack of the charge buff grants 218.6 DPS.
• A single death of a healer drops total raid DPS by 1748.8 DPS, while a single death of a dps drops total raid DPS by 6230.2 DPS.
• While DPS continues dropping as additional dps die, its rate of decline is decreasing.

The given:

• Raid composition: 2 tanks, 7 healers, 16 dps.
• Raid positioning: 12.5 players on each side of Thaddius, each side having 8 dps.
• Thaddius' health: 27 600 000 HP.
• Enrage timer: 360 seconds (6 minutes)
• Average size of stacking group: 12.5 players.
• Average stack of charge buff among dps: 11.5 stacks. This gives a damage boost of 115% of average personal DPS.
• Healers do not dps, tanks do minimal damage (in this scenario, VERY minimal).

Detail:

The figure that will be used as the average personal DPS, fully buffed, is the minimum personal DPS needed to down Thaddius before the enrage. The total raid DPS is:

27 600 000/360 = 76 666.6666... DPS

The total personal DPS is:

7666.6666.../16 = 4791.6666... ~ 4700 DPS (attribute the extra 91.6666... DPS to the tanks :))

Total personal DPS is:

100% average DPS + 115% bonus DPS from charge buff (decimal 1.15) = 215% of average personal DPS (decimal 2.15)

The average personal DPS is:

4700/2.15 ~ 2186 DPS

That means the bonus DPS from the charge buff is:

4700 - 2186 = 2514 DPS

A 1-stack of the charge buff grants:

2514/11.5 ~ 218.6 DPS

Consider the death of a healer. That removes a 1-stack of the charge buff off the 8 dps on their side. So, the drop in DPS from that death is:

218.6 x 8 = 1748.8 DPS. That's about 524.6k damage over 5 minutes lost!

Consider the death of a dps. That removes the dps' total personal DPS plus a 1-stack of the charge buff from each of the 7 dps on their side. So, the drop in DPS from that death is:

7 x 218.6 + 4700 = 1530.2 + 4700 = 6230.2 DPS. That's about 1.87M damage over 5 minutes lost!

To observe the change in DPS using calculus, additional deaths will need to be considered in pairs (one dps from each group will drop total raid DPS by equal amounts) and we will assume that only the dps die. The change in total raid DPS from the first pair of deaths is:

-6230.2 x 2 = -12 460.4 DPS

Calculating the incremental changes in DPS from additional pairs of deaths, we get the following progression of values:

-12 460.4, -12 023.2, -11 586, -11 148.8, ...

This is in fact an arithmetic progression with common difference of 437.2 and first term of -12 460.4. Total DPS lost from n pairs of deaths will be the following sum of values:

S(n) = 218.6n^2 - 12 679n

Differentiating S(n) with respect to n:

dS(n)/dn = 437.2n - 12679. Since the first derivative is negative over the domain 0 < n < 8, S(n) will continue dropping.

Differentiating dS(n)/dn with respect to n:

d2S(n)/dn2 = 437.2. Since the second derivative is always positive, S(n) is decreasing at a decreasing rate.

We have the following trend:


Originally posted here.

Succeeding at lava waves

Flame Tsunami (lava wave) is that annoying ability Sartharion uses that can send you spinning in circles. It is very difficult (impossible if Shadron is up) to heal through. Fortunately, there is some predictability you can exploit.

A lava wave will always start at either the north or south side of the island, then sweep across the island to the opposite side. All lava waves originating from the same side will always have the same configuration of gaps. If coming from the north, there will be a gap in the centre of the island, while if coming from the south, there will be gaps in the east and west ends of the island.

Just observing one side when Raid Warning calls out "The lava surrounding Sartharion churns!" is enough to tell where you need to be. Guild convention states that Sartharion be tanked at the south-east corner of the island, thus it will be easiest to turn towards the southern side when needed. There are two courses of action which you will need to choose between:

1. If you see the lava wave's originating there (at the southern end), you need to be at either end of the island. As Sartharion occupies a large portion of the eastern end, the tank should stay in the eastern gap and the rest should be in the western gap. Of course, in this case, you will easily be able to judge where you need to go since you are observing the lava wave directly.
   
2. If you don't see the lava wave's originating there (instead originating at the northern end), the tank should hop onto one of the "stepping stone" islands (off the main island) and the rest of the raid should be in the centre of the main island. However, during any drake phases, the raid may find that the (stagnant) lava off the west end is closer than the centre of the island. Do not be afraid to jump in; it is only slightly over lukewarm.

Doing this, you will not need to use the A and D keys as much.

Originally posted here.

The First Post

Hi, and welcome to my first ever blog! I will be using it for the general purpose, that is, I will blog about anything that strikes my fancy.

I do intend the title of the blog to reflect the focus of its contents. While I will be reporting facts, the main focus in all my posts here will be in extracting meaning from them, leading the post to implications, alternate perspectives, etc. You may notice that the title is drawn from the NSW (Australia) Board of Studies list of standard key words, which they refer to when drafting exam papers. This is the intended interpretation of the phrase, if you were curious.

While this blog is general purpose, I do expect to spend significant hot air and brainwaves on posts related to hobbies. My current main hobby as of time of posting is playing Blizzard Entertainment's fantasy MMORPG World of Warcraft. I have also played RuneScape and Neopets, but I do not actively maintain my characters there as of posting. Of course, this does not inhibit the potential for analysis in their respects.

As a minor note, I also intend this blog to be a more permanent location for my contributions to my WoW guild Altitude of Frostmourne US. An advantage of relocating here is that I can receive constructive scrutiny on my analyses from a larger audience. In turn, the contents of this blog may even be helpful to persons external to both the guild and even WoW.

I am starting this blog with high enthusiasm. I am hopeful that it will persist.