Mathcraft – building an arch

 

Buildings in Minecraft can actually follow certain shapes or patterns. Making an arch in Minecraft can be a challenge if you don’t know what your doing. Luckily, the quadratic equation can help map out where to place the blocks of an arch so that you can build something that will blow away those who see your creation.

Arches are commonly used for grand entrances. Using this in your build can create a new level of awe. Take a look at the picture below to get an idea of what you will be making.

The act of actually creating one that looks good may seem challenging. The quadratic equation can be used to map an arch. There are a few forms of the quadratic equation, but we will be using one that looks like this:

A(X-K)²+C=Y

This is essentially a slightly modified standard form. This may look a tad confusing since there are more letters and symbols than numbers, but it will make sense later on.

Prerequisites

There are few things you need to know before you begin building.

• How high you want to arch to go
• How long it needs to be

There is also some planning you will need to do. The arch will look better the wider and taller it is. A very thin or tall arch will look odd. A short but long arch would work well as a broad entrance into something, but it is important to know that something too short will cause problems for players walking through. Because of this, you need to make sure that your arch is somewhere in the middle in terms of size. I would recommend testing it out on creative so that you can ensure that the arch will look the way you want. If the formula below only gives you arches that don’t look the way you want, you will likely need to make some modifications to the build.

 

Getting the equation

Now it is time to jump head first into the math involved. I will try to explain everything as simply as possible. If anything goes wrong or you get stuck, just post a comment and I will try to help you out the best I can. The equation above isn’t usable yet. Instead, we need to break everything down and work on finding a number for each letter:

1. Ignore the equation above temporarily and instead think of this one
   1. A=-H/M².
   2. The variables (A, H, and M) all have specific meanings and will represent numbers. The variable H is the height of you build. You should have decided this part in the last section. The variable M is half of the width you decided earlier. If M is an odd number, you will have a decimal.

2. If you didn’t do this already, change the variables to their corresponding numbers. You will still have the variable A, but everything else should be numbers now. Here is an example if you had a height of 30 and wanted the width to be 10: A=-30/5²
3. Since the right side is entirely numbers, you can simplify it down. Using the example from before, I will end up with A=-1.2. As you see, there is a pretty high chance that you won’t have a nice whole number. I would highly recommend using a calculator. Some of you may not have a calculator sitting around near you. You can Google “calculator” to get a cool one in your browser. If you end up with some super long decimal, just keep calm and round.
4. We now have a number that equals the variable A. If you remember in the previous equation (A*(X-K)²+C=Y) you have a variable named A. All you have to do is replace the number you got in the last step with the letter A. Using the exact same example from earlier, I would have, -1.2*(X-K)²+C=Y.
5. Now that you have one of the letters removed, we need to get rid of a second one. take a look at the variable C. All you have to do is replace it with the height of the arch. Once again using the previous example, I would have, -1.2*(X-K)²+30=Y.
6. Time to change the K to a number. Do you remember half of your width? Replace K with that number. In my example, I would have, -1.2*(X-5)²+30=Y.

Good job making it so far! Hopefully it hasn’t been too overwhelming. If you are ready, continue onto the next few steps. Things will be much easier from here.

 

Using the equation

 

Everything you have done so far involved getting the equation to a usable state. There are currently two variables left. They are X and Y. These are special variables, as many of you who have taken algebra know, and the numbers they represent will change. The X variable is known as the input. Whatever you put into the input gets modified and comes out as the output. For us, the input will be equal to the distance we are horizontally from the left edge of the arch. The left edge will be an input of 0. The Y variable is know as the output. For us, this is the height of the block to place. With this information, it is time to start building.

1. Now you can start building. Start but replacing X with 0. Solve the equation’s left side. there should be only numbers.
2. You now have the output. Since you put in the input 0, you will be 0 blocks from the left edge. From the example I have been doing, I will have 0=Y. Hopefully you have the same thing. If not, something went wrong and you should check your work again or ask in the comments for help. Place the block 0 blocks from the left edge and 0 blocks up. This will always be your first block.
3. Now that you know how to find what blocks to place, you can do the rest. Just add 1 to the previous input and get the output. the next inputs will be 1,2,3,4… and the outputs will change based on the formula. Remember, just move over 1 block for each input and place the block at the height given by the output.

Congratulations! You now know how to make the perfect arch. At the bottom you can see what I made using the the example I have been carrying throughout.

The example arch

Here is a table of values. These are the inputs and outputs from the example.

A table of values for the example

x values (inputs)	Y values (outputs)
0	0
1	10.8
2	19.2
3	25.2
4	28.8
5	30
6	28.8
7	25.2
8	19.2
9	10.8
10	0

Finally, I have the finished product. This is just like the one at the beginning, the only difference is that it doesn’t contain any of the extra decorations.

If you want another arch example, take a look at the one below.

I hope this tutorial helps you make your next build even better with the use of arches.

If you enjoyed this post, check out my other Mathcraft tutorial here.