Elites of Minecraft: The Architect

ELITES OF MINECRAFT
The Architect
by Featherblade

Contents

The Tools of the Architect

The tools of the architect are the means of which allow you to create perfect buildings and curves, and are essential to learn. Many of these tools use simple software like MSpaint or Google calculator. Some more advanced things (such as accurate parabola curves) will require a scientific calculator, which may or may not be unobtainable. Alternatives will be offered to a scientific calculator if possible.

These tools can be used singularly or in conjunction with each other to create flowing artistry and permit variance even if the same tools are repeated over and over. Without these tools, the Architect has nothing to build with but his own ingenuity: essentially, the guide is useless! So take time to learn these tools, and run through them when you come to a choice in the building construction.


The Golden Ratio

The golden ratio is the perfect ratio between one thing and another. It's definition is a tad obscure, where: [1]


This image is straight from wikipedia. Essentially, if you times or divide any length by the golden ratio, you will get a length which is in perfection with the golden ratio. This perfection creates an extremely appealing look, which is used in artistry and in nature.

So, here are some examples of golden ratio usage:

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As a transition for different materials.

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As the exterior corner of a building.

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As the interior corner of a building

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As a change in depth. *Bonus* Make the total change in depth in golden ratio to the length!

Using these variations, we can plan out the raw size of a building constrained within a golden ratio cuboid, or made of golden ratios itself. This is the main usage of ratios: as a line in part of a shape.

Calculating New Lengths

So you are ready to start planning something out with golden ratios, like a cuboid or a floor plan. However, you might not be sure where to start. There are a few lengths that divide and times very easily with the golden ratio: they are the Fibonacci numbers.

Rather than explain the numbers themselves, I'll just tell you what they are. They will be explained in the Fibonacci section.

These are the numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233


These numbers are the special dimensions which should be used when plotting out floors and the height of walls. Every number on the line is a perfect golden ratio with the number to the left and the right of it. For example, '3' is in golden ratio with both '5' and '2'. Pairing up golden numbers in a building is the best way to use the golden ratio.

Now you should pick two numbers to create a rectangle. Remember to make sure the numbers are golden to each other (like 5 and 8). These two numbers can be arranged in any of the ways above. (For instance you can make an exterior or interior corner to create a rectangle).

Now I better tell you what the golden ratio actually is. As an irrational number, it is 1.618 : 1 .E.g, The next number in the sequence is 1.618 times bigger than the last number. The perfect golden ratio however is the formula ((1+sqrt 5)/2) : 1.

If you calculate (google or type this into your calculator), 5*((1+sqrt 5)/2), you will get the rough answer '8'. This is because as shown on the Fibonacci number line, 8 is just after 5. Likewise, 5/((1+sqrt 5)/2). you will get '3'. This works with any number.

If you type 6*((1+sqrt 5)/2), you will get roughly 10. It's not quite 10, so it's not quite perfect in Minecraft. As such, try and stick to the Fibonacci numbers if possible, as they are all relatively whole numbers.


I talk about the Golden Ratio in the video below. Specifically, I talk about how to derive lengths for secondary rooms and heights using the golden ratio. Just a note - I hate telephones.

One key thing to remember is that points of interest appear at each corner: The height may not be golden ratio to one side, but it can be to another off the same corner. This can results in four different options to choose from when deriving lengths.

For example, lets use 5 and eight. We have a five by eight rectangle, and want to make the height for it to create a cuboid. At any corner of our rectangle, we have the sides 8 and 5, so we can use any of those for our golden ratio height. This leaves us with:

8/((1+sqrt 5)/2) or 5 8*((1+sqrt 5)/2) or 13 5/((1+sqrt 5)/2) or 3 5*((1+sqrt 5)/2) or 8


Because of this, we have a range of heights for our cuboid, which leaves a range of options to choose from.

lTSYl.png

Using the same example, this is a spectrum of possible heights. Because all of these sizes result in four of the six sides of the cuboid being in golden ratio, you should choose out of purpose more than anything. An office or hotel would be very high, whereas a house would be lower and a platform lower still.


When you have multiple golden rectangles, you should make the most of the faces on the cubes. If we wanted the height the same across both rectangles, we should find out the maximum number of golden ratio rectangles we can make the with height.

r0Loh.png

This cuboid has 6 faces which can be made golden ratios without midpoints. These sides are lengths 3, 3, 5, 5, 8 and 8. To find the best height for all of them, we can use the series of golden ratio times and divides on all lengths and see what number comes up the most.

8/((1+sqrt 5)/2) or 5 8*((1+sqrt 5)/2) or 13 5/((1+sqrt 5)/2) or 3 5*((1+sqrt 5)/2) or 8 3/((1+sqrt 5)/2) or 2 3*((1+sqrt 5)/2) or 5


So we got the numbers 2, 3, 5, 5, 8 and 13. Because '5' appears more than any other, it is the best number golden ratio wise to choose. 4 of the 6 sides will be in golden ratio with a '5' height. Providing that a height of '5' allows the purpose of the building to be carried out, it should be used. If that number is too tall or too short, another height can be chosen, but note that it won't look as good...

Using Midpoints

Not always can you just make a house out of golden ratio cuboids. Instead, subtle use of midpoints can add even more appeal to a construction. Essentially, midpoints locate key features as a point within golden ratio sides. These include doors, windows, pillars, a change in texture or colour, the start of a new room and more.

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This here is a mid-point. Mid-points break up bigger golden ratios into smaller ones, making the structure more appealing. Because the mid-point is not often an actual block, but a point between two blocks, it can be slightly hard to add features to a mid-point.

For example, here is a small house with two types of midpoints.

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This house uses the door as an interior trim, creating a 5x3 rectangle on the right, while the window has it's centre right over another mid-point, creating another 5x3 on the left. To find out where the midpoints are, take the length of the side you want to add mid points to, and divide it by the golden ratio. The resulting number is the appropriate distance out from either side to create one or two midpoints.

As a transition between two materials, you must coinsider whether to use an 'interior trim', 'exterior trim' or 'dual trim'. This involves deciding which side contains the actual barrier between the materials. Here are some examples of the various styles of trim.


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No trim. These tend to look sudden in change and not appealing to look at. Avoid these.


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Interior trim. This pushes the golden ratio back (unless the trim material is very similar to the main material). These should also be avoided if possible.


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Exterior trim. The line is located outside the main golden ratio, bordering it. This preserves the golden ratios, so is good to use. It's also very subtle.


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Dual trim. The two lines provide a definite barrier between the change in materials. The barrier is balanced on both sides, and provides more variety to the building. These are the most optimum trims, although they tend to require more room to make properly.


Midpoints also make excellent diversion points to create more rooms from. Typically, if you use golden ratios often in your builds, you will notice that a straight line across two golden ratios will create a midpoint. Similarly, interior corners made by golden floor plans will also be a form of midpoints.

Golden Shapes

There are numerous golden shapes, which are regular shapes with the golden dimensions. These shapes start off simple, but soon become shapes that require too much definition to work in minecraft. These shapes include:

  • Rectangle
  • Ellipse
  • Rhombus
  • Spiral


Rectangle


Up to this point, all the work on golden ratios were for rectangles and lines. Rectangles are the easiest to make, and tend to be the shape most commonly used in buildings.

Ellipse


Using paint, two numbers (one in golden ratio to the other) can be used for the height and width of an ellipse. This will create a perfect oval shape. These ovals are useful for large gates, the bases of towers or for the curve of a wall.

Rhombus


A golden rhombus is a diamond shape bound within a golden rectangle. Either the total height versus the total width or vice versa will need to be in golden ratio with each other. This can again be plotted out in paint using lines.

Spiral


[2]

As the graphic shows (from wikipedia), a quarter circle is drawn from the 'shorter' side corner of the rectangle, to a golden midpoint. When a line is drawn perpendicular from the midpoint, another, smaller golden ratio rectangle is created. This goes into infinity smaller and smaller.


The spiral is about as complicated as you can get in minecraft. Other shapes, such as pentagons, pentagrams and triangles are able to follow golden ratios, but to a scale unachievable in minecraft, thus do not need to be mentioned.

Parabolas

Curves are very important in a world that has no naturally existing ones. The parabola is just a simple curve that can be used to create a realistic effect on structures like bridges and towers. The reason people use parabolas is because they share weight perfectly along then bulk of the structure and into the supports.

Parabolas have a surprising range of uses, right from the under trusses of a bridge, to the cabling of a suspension bridge, to making the bridge an arc itself. In towers and buildings, materials can protrude from the wall in parabolic fashion, and arches can be constructed within cathedrals in the form of archways and high dome roofs.


To create a parabola, you will need a scientific calculator with a table function. If you don't have a table function, substituting the 'X's in the formulae would be ok, but would require a ton of work. In it's basic form, Y=aX(X-b) will be used in this guide. 'Y' is the height of the parabola at every point of 'X', where 'b' is the size of the parabola and 'a' is the scaling factor.

To work out a parabola, you must first enter two digits: The height (Y) and length (B). Because a parabola is symmetrical, the highest point (Y) will always be directly halfway on the parabola, so 'X' = half of 'b'.

An example is shown here. We want a twenty wide, five high parabola:

Y=aX(X-b) 5=a10(10-20)


Before we go any furthur and find out 'a', there is an important note. When working with parabolas, you are working with co-ordinates, not blocks. What I am actually drawing here is twenty points, with nineteen spaces between them. As such, for accurate readings, always add one to your width. So re writing the example:

Y=aX(C-b) 5=a10.5(10.5-21)


Next, you need to find out 'a'. This can be done by simplifying the equation.

5=a10.5(10.5-21) 5=a10.5(-10.5) 10.5*-10.5=-110.25 5=a*-110.25


Lastly, divide the height by the resulting number. This will leave you with the scaling factor 'a'.

5=a*-110.25 5/-110.25=-0.04535... 'a' = -0.04535...


Once you have the scaling factor, open up a table on your calculator. You will need to retype the original equation again, but without height and width. The calculator will then list the heights for the width.

Y=aX(X-b) Y=-0.045X(X-21)


This is just the basic working out. The three main types of parabolas (simple, half and inverted) each have a different formula from each other. Still, this is how you work the equations, no matter how complex they get.

Note: If you lack a scientific calculator, you can plot the parabola online by entering the formula into WolframAlpha.

Simple Parabolas

A simple parabola is a upward arch, and is used the most by structures. These arches start and end at points along the same height, which make for fine bridges. There are three general types of simple parabolas, providing different walking surfaces/tops. All these arches are created with the standard Y=aX(X-b).

Simple Basic Parabola

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A basic parabola is rounded to the nearest whole block for simplicity. When reading the numbers on the chart, 0.5-1 are rounded up, and 0.5-0 are rounded down. As such, you will get the parabola's height to the nearest whole block.

Whilst this is not particularly accurate in small cases, this parabola can be used for building arches on larger structures where people are not expected to walk over, and can be built as the roofs of cathedrals or town halls and the like.

Simple Accurate Parabola

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An accurate parabola boosts added definition at the same size of simple parabolas. When creating the parabola, round 0-0.25 down, round 0.25-0.75 to a half and round 0.75-1 up. This results in far more accuracy than any other type of simple parabola.

What is useful about these is that the half steps make the structure walk-able if the height-width ratio persists, making it very useful for constructing bridges. Unfortunately, half blocks may only be placed on the lower portion of a block, so accurate parabolas must be constructed upright, and cannot be accurate on the underside.

Simple Exterior Parabola

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These parabolas are unique in that they can provide a flat top. These are very useful for bridges with which you don't want a change in elevation of the walking path whatsoever. There are two ways to make these: Either create a parabola and fill in the space above it, or you can then remove the original parabola itself.

Half Parabolas

The half parabola is a neat way of making a curved bridge across two points with different heights. The actual formula for working these out is the same Y=aX(X-b), but the 'X' value now stands for the width plus 0.5 (co-ordinate issue) and 'b' now stands for double the width plus 1 (again, the co-ordinate issue).

There are only the basic and accurate types for half parabolas. Having a flat surface is mute when you want a change in height.

Half Basic Parabola

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This parabola has the advantage of only spanning one way. This is great for if you are making foundations for large skyscrapers or supports from a building to supporting buildings or terrain. The most obvious advantage is a flowing curve across multiple heights.

Half Accurate Parabola

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This parabola is very useful in the construction of sky bridges, due to the ease for changing height between structures and for it's walk-able nature. It is created by rounding 0-0.25 down, 0.25-0.75 to a half, and 0.75-1 to a whole.

Inverted Parabolas

Unlike the other techniques, Inverted parabolas are the only upside down parabola style. By going from two high places of the same height down into a dip, things like cables and supports are possible. These inverted parabolas can be used in conjuction with others to create very realistic or flowing buildings and bridges.

The formula for calculating inverted parabolas is a little different. The formula Y=-aX(X-b)+Y is used. Essentially, you are flipping the parabola (inverting it) with the negative 'a', and then raising it back into positive numbers with the + 'Y'.

Inverted parabolas can exist as basic or accurate:

Basic Inverted Parabola

~Image needed~

The parabolas create awesome flowing effects for cables and ties, making them great to use for air-ground anchoring.

Accurate Inverted Parabola

~Image needed~

These parabolas tend to have an astounding effect, mainly because cables tend to be large and the accuracy creates flow effects never before seen in minecraft.

Angled Perspective

Minecraft is a game of blocks. Metre by metre blocks. Because of this, we tend to forget about the length of lines at angles. Some great mistakes making polygons in minecraft can be fixed by using scaling factors to change an actual distance (like a diagonal) into a minecraft horizontal distance.

Here is an example of using (and not using) scaling factors for angles.

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1: The rectangle on the left is a 8 by 5 golden ratio rectangle. This is what I have attempted to replicate at a 45 degree angle. 2: The middle rectangle is a correct translation between 0 and 45 degrees. Note how there are four blocks instead of five: this is because the actual perspective length is longer. 3: An incorrect translation of diagonals. By directly copying the five and eight onto a 45 degree angle if have made the rectangle LONGER than it should be.

Circles

Minecraft is completely made up of voxels, or cubes. When building something it is helpful to make it look as minimally blocky as you can, and you can do this with organic curvature, parabolas,

and also circles, which are an obvious decision. Circles can be done two ways - the good way, or the bad way. Here the good way will be shown and explained and at the end there will be an example

of a circle that has been made using different rules and doesn't appear as aesthetically pleasing.

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The first thing to do when creating a circle is to decide the approximate size. Here is a 3 blocks long white line. The circle made at the end of this will be a little longer than 3 blocks,

depending on how the curves are set up. This first number, also known as the 'side number' is the most important and for the easiest and best looking circles, the side number should be an

odd number in its length.

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The next part of building your circle is to decide how to create your 'connecters', in this case, the parts that are in blue. These are the parts of your circle that join up the

four side lengths, and are also extremely important as they are what makes the circle not a square. In the picture above the connecters are 1 block long. Later through this thread

there will be a collection of almost all the connecters you will ever need to use.

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This step is simple. Get the original side length, and copy it over three times, also copying the connecters to give your circle a complete perimeter.

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Here is an example of a circle with an even side length (on the left), and a circle with an odd side length (on the right). You can probably see how the one on the left appears better. (Warning: Complex bit approaching) This is because of the relationship

between the overall size of the side lengths and the overall size of the connecters on the left one are in a closely-related scenario. To be exact, the side length is only double the size

of the connecters, making it appear slightly more diamond-like than circle-like. This diamond-like property will increase severely as the size of the circle reduces.

50DRU.png

Here are the main, most common, and most useful connecters for circles. Please note that minecraft is a game without rules, and when you are calculating your circles it is good to experiment

with different side lengths and different connecters until it appears just right.

FeLQ3.png

Why is this not a circle? Well, here is why:

Notice that the side lengths, which have been placed in a red colour, are one block long, and also notice how each of the connecters are one block long (They are one block long connecters,

placed three in a diagonal row). As explained earlier, when side lengths and connecters have small or no differences, it appears like a rhombus or diamond rather than a circle.


Material Grain

Although using aesthetically pleasing curvature and correct usage of circles in minecraft, an architect must not forget that it is not just how you place the blocks but also what the blocks

actually are. Observe this example of a wall segment:

Mm9b1.png

Although this wall uses a nice selection of blocks, they are not quite in the right places. For example, the double-slab blocks surrounding the window have horizontal line alignment and should

be used in long, horizontal places rather than as an outline. And this is the same with the pillar blocks in the corners, as they have vertical line alignment so should be placed in vertical, tall

areas to make the wall look better. Here is another wall using the same blocks, in better places:

1mJmb.png

As you can see, the vertical blocks are on the sides, and the horizontal aligned blocks are on the top. 'Neutral' blocks, in this case the wooden planks, are placed in neutral areas, serving as a fill,

and they can also be used as a corner block sometimes.



On The VoxelBox's YouTube videos, forums, and other sites and locations, there is a lot of messages to be found saying that players who teach others things like you have just seen are hypocritical

since they never use these methods themselves in their own builds. They do! Take a look around the builds, and you will usually be able to spot something like the golden ratio, or a correct parabola.

Some do not think these methods are used by their creators since the creators are so good at using them, it is invisibly integrated into the builds. We all hope you take these tips into consideration.

Change Log

The change log will record the history of the thread from release date.

  • 20/05/12: (Peridox) Added information on circles, with permission from Featherblade. Beginning of the continuation of this thread.
  • 16/12/10: Added some text to 1.21 to help clear confusion.
  • 16/12/10: Added change log, credits and changed an incorrect picture with the correct one.
  • 15/12/10: Initial Thread released.

Also see: Geometrical Curve Approach