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.

A little known trick off of the forums is the Click Mining Technique. I have often referred it to speed mining, but that is linked to 'clocking', a glitch in SMP Alpha. The Click Mine is not a hack tool, nor is it able to be considered 'unethical' as far as role-play goes.

Click Mining is a skill. One can use it to boost the speed of mining stone, ores and even dirt.

So, what is Click Mining? Click Mining is just what it is: You re-click after destroying each block. It's not that hard. After you destroy a block, let go of the mouse button and re-click as fast as possible. There are two ways to boost your speed like this:

Let go of the mouse sooner after destroying the block.

AND

Re-click faster after destroying the block.

If you do it right, you can shave off approximately 0.154-0.160 seconds off of your block* destruction!
*Tested with cobblestone and wood, approximation due to error in skill and stopping the timer.
...now, that doesn't seem much, but take a look at this table. (Credit to Crixinmix)

Take away 0.160 from Smooth Stone with an iron pickaxe, and you're almost exactly 30% faster! Do it with dirt, and you're a whooping 60% faster! (with iron shovel). This gives incredible destruction speed to any Miner, no matter the style of mine, or even if it's true mining. This skill still applies to logging, building and demolition in general.

]Now here is where skill plays it's cards: If you let go and re-click before the block is destroyed, you lose all progress on destroying it. This can waste that half second you just made up through speed mining. If you re-click too late, you will not gain any time on destroying the block with clicks over a hold. Also, you must be prepared to change how fast you click on stone, coal, iron, cobblestone and dirt with a range of tools to make maximum benefit with no down time.

All in all, this is a skill which when mastered, leads to far faster game play, and keeps you active when solid stone mining. When applied in conjunction to a efficiency mine, you can make yourself an Elite Miner fast.

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

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:

As a transition for different materials.

As the exterior cornor of a building.

As the interior cornor of a building

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 ratio's itself. This is the main usage of ratios: as a line in part of a shape.

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 cornor: 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.

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.

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...
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1.22| 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.

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.

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 'duel trim'. This involves deciding which side contains the actual barrier between the materials. Here are some examples of the various styles of trim.

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

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.

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.

Duel 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.
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1.23| 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

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.
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1.30| 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-:cool.gif: 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 (:cool.gif:. 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-:cool.gif:
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-:cool.gif:
5=a10.5(10.5-21)

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

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-:cool.gif:
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.
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1.31| 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-:cool.gif:.

Simple Basic Parabola

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

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

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.
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1.32| 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-:cool.gif:, 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

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

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.
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1.33| 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-:cool.gif:+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.

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.

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

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.
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Bookmarked. Very useful as quick reference when building (I never apply complicated geometry to my structures because I'm lazy =/ ). Well written, and straightforward. I think you are my favorite poster on this forum now, by the way. Consistent 5/5 threads. Thanks for doing so much for the community!

Useful, but it seems strange that you disproportionately limit a treatment of classical architectural theory... to a lesson about parabolas; they are just one of the conic sections... you did not give a mathematical overview of any of the others... or any other curver for that matter, lol. Why parabolas? Classical architecture is far more obsessive about circles, segements of circles, and the relationship of regular convex polygons... and circles.

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.

For instance, if you analyze the curve segments in Gothic elements, they are nearly without exception circular segements.

Useful, but it seems strange that you disproportionately limit a treatment of classical architectural theory... to a lesson about parabolas, lol. Why parabolas? Classical architecture is far more obsessive about circles.

This was mainly due to lack of time more than anything. This guide is ongoing: Circular and ellipse curves will be added. I chose parabolas because they can be made any shape or size, and usually gained a walking surface quality faster than any ellipse.

I also chose parabolas as they exist naturally in larger bridges such as suspension bridges, and even smaller arch truss bridges contain parabolas. I quiet honestly think circles are rather useless alone, and require backup from other tools of the architect, plus virtually everyone knows how to make a circle already.

They will be added in due time, which will probably be a bit later today. Also those angles will be clarified, with methods for obtaining any angle and scaling factor.

Quote from The_Pastmaster »

You totally lost me in 1.21

Which part exactly, the part where I talked mathematics about timesing and dividing the golden ratio? This is important, as the guide should be understandable.

Which part exactly, the part where I talked mathematics about timesing and dividing the golden ratio? This is important, as the guide should be understandable.

Right after the Fibonacci Number sequence. I feel like your talking over my head because I haven't a clue what your on about nor why it's useful.

Hooray for parabolas! I'm gonna give some of this "golden ratio" stuff a try. Also, could you add some pics of what finished golden ratio/parabolic creations look like? I'd like to see it in action.

The idea behind using Golden Ratios in architecture is that it is perceived as being more pleasing. Apparently.

Could you do me a favor and clean the math up? In the Golden Ratio section you seem to outright state that the ratio of any two consecutive fibbonacci numbers is the golden ratio, when it is really only an approximation. Later on you do recognize that it isn't perfect, but only after you've 'explained' the math. I understand that being precise under these circumstances is impossible, but rather than stating 'there is imprecision because of how Minecraft works' you make several statements that are outright false.

Could you do me a favor and clean the math up? In the Golden Ratio section you seem to outright state that the ratio of any two consecutive fibbonacci numbers is the golden ratio, when it is really only an approximation.

A ridiculously precise approximation at that. I took it out since the last edit, but I referenced that numbers 3 and up will accurately show the golden ratio. If you take the tools shown in the thread, you can do the math to find out how accurate the numbers are.

Take a number, 3. Times that by ((1+sqrt 5)/2. That will give you 4.854... That 14.6% of a missing block is equivalent to roughly two and a third pixels in game. It's not that much error, considering the limits of minecraft.

5 Timesed by ((1+sqrt 5)/2 will give you 8.090... That's less than 1% of a extra block.

8 Timesed by ((1+sqrt 5)/2 will give you 12.944... That's less than half a percent of a missing block.

13 Timesed by ((1+sqrt 5)/2 will give you 21.034... That's less than a third of a percent of a extra block.

The bigger you go, the more accuracy you get.

The reason the math isn't so perfect with non-fibo numbers is because they end up halfway through a block. There is 18% of an extra block when timesing 10 by the golden ratio as an example. Any number between the fibo numbers will be less perfect than the fibos themselves. It means you can still build a 12 by 19 room, but you won't be able to account for that extra half a block that should have been added.

And I will try to clear up the mathematics. Bits of it are obviously not clear.

It uses a lot of painterly textures, but some of the blocks not used in architecture (sponge, adminium, all the ores, pumpkin, ect) have been replaced with usable textures (like thatch, marble, carpet, rose granite, columns and more)

It looks horrible on RP servers (deposits of carpet anyone?) but it does make for neat structures, especially for modern and greco-roman areas.

Kinda complex on tue parabolas bit, but I got the Fibonacci numbers bit: basically, if you use two consecutive FNs, you get a room that our brain just likes the look of for mo apparent reason

Sooo... My brain likes a 3 by 5 room better then a 3 x 3?

Trust me, I've been trying to find out WHY it does for the last few months.

Polls conclude it works. When presented blank rectangles of different size ratios, the 1.62:1 came out on top (which was the closest to the ratio), and people like Leonardo da Vinci used the golden ratio in paintings: The mona lisa's face is bound within a golden ratio rectangle.

If anyone actually has an answer to WHY the golden ratio is so good, feel free to tell the world right here.

Polls conclude it works. When presented blank rectangles of different size ratios, the 1.62:1 came out on top (which was the closest to the ratio), and people like Leonardo da Vinci used the golden ratio in paintings: The mona lisa's face is bound within a golden ratio rectangle.

I don't like La Gioconda. It's not a bad paining but I wouldn't buy one and if someone gave it to me I would either sell it or toss it.

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Venit, quessit, induravit.

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The Architect~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

1.00| The Tools of the Architect

1.10| Click Mining

1.20| The Golden Ratio

1.21| Calculating New Lengths

1.22| Using Midpoints

1.23| Golden Shapes

1.30| Parabolas1.31| Simple Parabolas

1.32| Half Parabolas

1.33| Inverted Parabolas

1.40| Angled Perspective1.41| Calculation of True Angles

1.42| Uses in Shapes

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A little known trick off of the forums is the Click Mining Technique. I have often referred it to speed mining, but that is linked to 'clocking', a glitch in SMP Alpha. The Click Mine is not a hack tool, nor is it able to be considered 'unethical' as far as role-play goes.

Click Mining is a

skill. One can use it to boost the speed of mining stone, ores and even dirt.So, what is Click Mining? Click Mining is just what it is: You re-click after destroying each block. It's not that hard. After you destroy a block, let go of the mouse button and re-click as fast as possible. There are two ways to boost your speed like this:

Let go of the mouse sooner after destroying the block.

AND

Re-click faster after destroying the block.

If you do it right, you can shave off approximately 0.154-0.160 seconds off of your block* destruction!

*Tested with cobblestone and wood, approximation due to error in skill and stopping the timer.

...now, that doesn't seem much, but take a look at this table. (Credit to Crixinmix)

Take away 0.160 from Smooth Stone with an iron pickaxe, and you're almost exactly 30% faster! Do it with dirt, and you're a whooping 60% faster! (with iron shovel). This gives incredible destruction speed to any Miner, no matter the style of mine, or even if it's true mining. This skill still applies to logging, building and demolition in general.

]Now here is where skill plays it's cards: If you let go and re-click before the block is destroyed, you lose all progress on destroying it. This can waste that half second you just made up through speed mining. If you re-click too late, you will not gain any time on destroying the block with clicks over a hold. Also, you must be prepared to change how fast you click on stone, coal, iron, cobblestone and dirt with a range of tools to make maximum benefit with no down time.

All in all, this is a skill which when mastered, leads to far faster game play, and keeps you active when solid stone mining. When applied in conjunction to a efficiency mine, you can make yourself an Elite Miner fast.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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

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:

As a transition for different materials.As the exterior cornor of a building.As the interior cornor of a buildingAs 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 ratio's itself. This is the main usage of ratios: as a line in part of a shape.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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:

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 cornor: 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 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.

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.

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 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...

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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.

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.

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 'duel trim'. This involves deciding which side contains the actual barrier between the materials. Here are some examples of the various styles of trim.

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

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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

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.

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.

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.

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.

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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-:cool.gif: 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 (:cool.gif:. 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:

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:

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)

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/-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=-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.

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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-:cool.gif:.

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.

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.

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.

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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-:cool.gif:, 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.

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.

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.

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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-:cool.gif:+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:

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

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.

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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.

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.

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

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.

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A simple suggestion on geology here.

~~~

Slaves of the Coal Mine

An interesting Novel to pass the time.

I have a new signature as that one was a year (or more?) outdated!

+(insert large number here)

For instance, if you analyze the curve segments in Gothic elements, they are nearly without exception circular segements.

Massive Greek Temple

Greatest Waterslide Ever Built?

The Great Labyrinth

Classical Portal Temple and Piazza Project

Great Maze Making Contest

Venit, quessit, induravit.This was mainly due to lack of time more than anything. This guide is ongoing: Circular and ellipse curves will be added. I chose parabolas because they can be made any shape or size, and usually gained a walking surface quality faster than any ellipse.

I also chose parabolas as they exist naturally in larger bridges such as suspension bridges, and even smaller arch truss bridges contain parabolas. I quiet honestly think circles are rather useless alone, and require backup from other tools of the architect, plus virtually everyone knows how to make a circle already.

They will be added in due time, which will probably be a bit later today. Also those angles will be clarified, with methods for obtaining any angle and scaling factor.

Which part exactly, the part where I talked mathematics about timesing and dividing the golden ratio? This is important, as the guide should be understandable.

A simple suggestion on geology here.

~~~

Slaves of the Coal Mine

An interesting Novel to pass the time.

Right after the Fibonacci Number sequence. I feel like your talking over my head because I haven't a clue what your on about nor why it's useful.

Venit, quessit, induravit.I will now use this knowledge to rule the world!

Could you do me a favor and clean the math up? In the Golden Ratio section you seem to outright state that the ratio of any two consecutive fibbonacci numbers is the golden ratio, when it is really only an approximation. Later on you do recognize that it isn't perfect, but only after you've 'explained' the math. I understand that being precise under these circumstances is impossible, but rather than stating 'there is imprecision because of how Minecraft works' you make several statements that are outright false.

A ridiculously precise approximation at that. I took it out since the last edit, but I referenced that numbers 3 and up will accurately show the golden ratio. If you take the tools shown in the thread, you can do the math to find out how accurate the numbers are.

Take a number, 3. Times that by ((1+sqrt 5)/2. That will give you 4.854... That 14.6% of a missing block is equivalent to roughly two and a third pixels in game. It's not that much error, considering the limits of minecraft.

5 Timesed by ((1+sqrt 5)/2 will give you 8.090... That's less than 1% of a extra block.

8 Timesed by ((1+sqrt 5)/2 will give you 12.944... That's less than half a percent of a missing block.

13 Timesed by ((1+sqrt 5)/2 will give you 21.034... That's less than a third of a percent of a extra block.

The bigger you go, the more accuracy you get.

The reason the math isn't so perfect with non-fibo numbers is because they end up halfway through a block. There is 18% of an extra block when timesing 10 by the golden ratio as an example. Any number between the fibo numbers will be less perfect than the fibos themselves. It means you can still build a 12 by 19 room, but you won't be able to account for that extra half a block that should have been added.

And I will try to clear up the mathematics. Bits of it are obviously not clear.

A simple suggestion on geology here.

~~~

Slaves of the Coal Mine

An interesting Novel to pass the time.

Oh yeah, and what texture pack are you using? ._.

It uses a lot of painterly textures, but some of the blocks not used in architecture (sponge, adminium, all the ores, pumpkin, ect) have been replaced with usable textures (like thatch, marble, carpet, rose granite, columns and more)

It looks horrible on RP servers (deposits of carpet anyone?) but it does make for neat structures, especially for modern and greco-roman areas.

Thank Plusnine for the compilation.

A simple suggestion on geology here.

~~~

Slaves of the Coal Mine

An interesting Novel to pass the time.

Sooo... My brain likes a 3 by 5 room better then a 3 x 3?

Venit, quessit, induravit.Polls conclude it works. When presented blank rectangles of different size ratios, the 1.62:1 came out on top (which was the closest to the ratio), and people like Leonardo da Vinci used the golden ratio in paintings: The mona lisa's face is bound within a golden ratio rectangle.

If anyone actually has an answer to WHY the golden ratio is so good, feel free to tell the world right here.

A simple suggestion on geology here.

~~~

Slaves of the Coal Mine

An interesting Novel to pass the time.

here have some cake

I don't like La Gioconda. It's not a bad paining but I wouldn't buy one and if someone gave it to me I would either sell it or toss it.

Venit, quessit, induravit.