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Fractals of the Mists are a special type of dungeon that consists of an array of mini-dungeons called fractals, where each fractal has its own story and environment. Characters are adjusted to level 80 within the fractal. Fractals are one of the primary sources of ascended equipment and provide materials for crafting infused equipment and attuned equipment. They are one of the core pillars of Guild Wars 2's endgame.

Fractals have some unique mechanics and design. The party can choose the difficulty scale prior to entering the fractal or from within the observatory. This difficulty scale starts at 1 and can be increased, up to 100. The personal fractal level is an account-wide stat that tracks the current position of the account on the difficulty scale. To increase the personal fractal level, and therefore the highest fractal scale accessible to the account, the player must complete a single fractal on a fractal scale equal or greater to their personal fractal level. This provides progression, allowing players to continuously complete higher and higher levels on the difficulty scale and earn greater rewards as a result.

When entering Fractals of the Mists, the players arrive in Mistlock Observatory, a \"hub\" area within the Mists. Entering Mistlock, you'll meet Dessa, a scientist who has made explorable chunks of reality from the Mists. There are hostile elements that were acquired during Mistlock which you are asked to dispose of. You may begin by entering the portal within the Mistlock to enter a fractal. You will hear Dessa's voice while stabilizing the fractal. Once it is stabilized, she and her krewe will transport you back to the Mistlock.

The Mistlock Observatory serves as the hub (or lobby) area for the dungeon. It offers armor repairs and a variety of merchants that sell fractal related gear, potions, and infusions. A portal in the center of the area can be used to access the fractals; this requires confirmation from all players in the party.

Fractal Attunements is a special mastery track introduced with Heart of Thorns. In addition to providing access to additional rewards and daily achievements, these masteries unlock certain NPCs in the Mistlock Observatory. Through these NPCs players can gain access to Infusion Extraction Devices, special crafting materials and other rewards. These are extremely helpful in completing the higher difficulty scales. In addition, players pursuing the Legendary Backpiece collection will need high fractal mastery to obtain the rare collection items required.

There are three types of fractal potions available: offensive (increased outgoing damage), defensive (decreased incoming damage) and mobility (increased movement speed) potions. These consumables stack up to five times (Large potions apply all five stacks in one use), and can be used in addition to nourishment and utility consumables. Depending on the Fractal Attunement mastery rank achieved, each potion may provide additional bonuses to vitality, toughness and concentration, respectively.

Account Augmentations are upgrades that can be purchased from Deroir and grant additional rewards for completing a fractal and account wide agony resistance. Each upgrade has four tiers and get progressively more expensive.

Each difficulty scale corresponds with a specific fractal 'island'. Agony is introduced at level 20, and its damage increases with the scale. The table below shows the island for each scale plus the minimum amount of agony resistance required to reduce damage from agony to its minimum of 1% of your total health per tick.

There are two types of daily fractals: Tier N and Recommended. In every tier there is at least one fractal for each of the three Tier N Dailies. In each of the three lower tiers (1-25, 26-50, 51-75) there will be one Recommended fractal.

Completing a Recommended fractal will reward players with a Research Chest depending on the tier. All Research Chests will give a Fractal Research Page and a large Fractal Potion depending on the chest.

Certain crafting materials can be obtained as random loot from foes in the Fractals of the Mists; unlike the above rewards, these drop from killing ordinary enemies, not for completing fractals. Each has a minimum difficulty scale at which it can appear. As of July 26th 2016, these can also be bought from INFUZ-5959.

In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set.[1][2][3][4] This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called affine self-similar.[5] Fractal geometry lies within the mathematical branch of measure theory.

One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radius of a filled sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the conventional dimension of the filled sphere). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer and is in general greater than its conventional dimension.[1] This power is called the fractal dimension of the geometric object, to distinguish it from the conventional dimension (which is formally called the topological dimension).[6]

Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass,[7] and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century.[8][9]

The consensus among mathematicians is that theoretical fractals are infinitely self-similar iterated and detailed mathematical constructs, of which many examples have been formulated and studied.[1][2][3] Fractals are not limited to geometric patterns, but can also describe processes in time.[5][4][13][14][15][16] Fractal patterns with various degrees of self-similarity have been rendered or studied in visual, physical, and aural media[17] and found in nature,[18][19][20][21] technology,[22][23][24][25] art,[26][27] architecture[28] and law.[29] Fractals are of particular relevance in the field of chaos theory because they show up in the geometric depictions of most chaotic processes (typically either as attractors or as boundaries between basins of attraction).[30]

The term \"fractal\" was coined by the mathematician Benoît Mandelbrot in 1975.[31] Mandelbrot based it on the Latin frāctus, meaning \"broken\" or \"fractured\", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.[1][32][33]

The word \"fractal\" often has different connotations for the lay public as opposed to mathematicians, where the public is more likely to be familiar with fractal art than the mathematical concept. The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with a little mathematical background.

This idea of being detailed relates to another feature that can be understood without much mathematical background: Having a fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are usually perceived. A straight line, for instance, is conventionally understood to be one-dimensional; if such a figure is rep-tiled into pieces each 1/3 the length of the original, then there are always three equal pieces. A solid square is understood to be two-dimensional; if such a figure is rep-tiled into pieces each scaled down by a factor of 1/3 in both dimensions, there are a total of 32 = 9 pieces.

We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/r, there are a total of rn pieces. Now, consider the Koch curve. It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3. So, strictly by analogy, we can consider the \"dimension\" of the Koch curve as being the unique real number D that satisfies 3D = 4. This number is called the fractal dimension of the Koch curve; it is not the conventionally perceived dimension of a curve. In general, a key property of fractals is that the fractal dimension differs from the conventionally understood dimension (formally called the topological dimension).

This also leads to understanding a third feature, that fractals as mathematical equations are \"nowhere differentiable\". In a concrete sense, this means fractals cannot be measured in traditional ways.[1][4][34] To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of measuring with a tape measure. But in measuring an infinitely \"wiggly\" fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. The result is that one must need infinite tape to perfectly cover the entire curve, i.e. the snowflake has an infinite perimeter.[1] 153554b96e

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