How high could a Lego tower be?

Question

I just won the national lottery! I am a billionaire, and can do whatever I want with my money!

Tv interviewer asks me what would I do with this money. Then, come the Idea.

I remember this guy when I was young, Kevin, if I remember correctly. His parents were rich, way richer than mine. And therefore, he had much more Lego than me. As he was my direct neighbour, he used to play in the garden, or to expose his big towers on her balcony, so I could see that he had more Legos than me.

I still remember him, and want my revenge.

Here is my plan. Kevin lives in a rural area, and there are only fields across the roads. I buy them all. Now, I want to build the highest Lego tower possible, just in front of his house. Money isn't a problem anymore, the sky is the limit.

• I would be ok if the foundation are not in Lego, but otherwise, all the towers must be simple Lego bricks.
• width/length shouldn't be a problem, as I have a big land

How high could my Lego tower be?

PS: This have been asked/answer elsewhere, but I'm not ok with the answers. For example:

• Quora: Just tell us a random number, without showing us the calculations. Bonus point for wondering if we have enough pieces.
• BBC: Base theorical heigh according to pressure/weight until a brick collapse, before telling us it would be impossible

Other articles are similar, based on the weight, but don't include the need of foundations, wind resistance...

Answer 1

10.8km high.

I wrote up a little program (code below) to find the exact break point at which we reach the 4240 Newton limit that was provided as enough to crush a lego block by your BBC article. I picked a pyramid, an extremely stable structure, and one that has a very wide base to properly distribute the weight above it, allowing it to get much taller than a tower. I hope you really do have plenty of land! I am running off of the following assumptions, that or course aren't necessarily true:

1. Your legos are being built on a perfectly flat surface.
2. The legos distribute their weight perfectly

Results

1125001 lego blocks high. 1125001 * 9.6mm ~= 10.8km
You could get this result for a while, bot note that your tower will slowly sink down over time, plastic deformation is bound to happen! You now have the pleasure of owning the biggest lego tower ever, but also the tallest man-made structure ever! Now just to watch it slowly melt...

Code

i = 1
weight = 0
totalLegos = 0
while True:
    # Size of new bottom layer
    base = i**2
    totalLegos += base
    # Add size of previously added layer to
    # total weight load on the base layer
    weight += ((i-1)**2) * .001152 # Weight of lego is 1.152g -> kg
    # Too heavy?
    if weight/base >= 432:
        print("Maximum reached!")
        print("{}/{} = {} >= 432".format(weight, base, weight/base))
        print("    Levels: {}".format(i))
        print("    Total weight: {}".format(weight+base*.001152))
        print("    Total legos:  {}".format(totalLegos))
        break
    else:
        print(i)
    i += 1

Output

546752187002808.0/1265629500004 = 432.00019200017067 >= 432
    Levels: 1125002
    Total weight: 546753645007992.0
    Total legos:  474612539069437505

Sources:
BBC
Exact lego dimensions