Shape of Compost Pile

I have been doing experiments with different shapes of compost piles. In geoff lawton's online PDC he suggests a gravity fall pile that ends up in a pyramid shape. Some long time biodynamic gardeners always do the trapezoid shape. I would love to hear opinions on this.

The igloo shape is easy to achieve and the most thermally efficient shape with a flat bottom. This might be important in a cooler climate. It's also a simple shape to cover with a tarp.

I agree with Dale, a pile that best approximates a sphere is the one that will generate and hold the most heat. That should correlate with faster composting.

Sometimes, when the pile is dry and there's rain forecast, I'll arrange the pile to collect and hold water, otherwise it's best to shed water, although rain water really makes it cook! Good luck!

Do you have a specific aim in mind? Breaking down animal bones? dealing with cooking oil? Just garden adn kitchen waste along with some gras/leaves/clippings?
If you need the bacterially hot pile for your aims? Is there an easier option?

My personal bias is toward messiness, to mimic nature.
I like the works and experience of Michael Phillips. (The Apple Grower, lost nation orchard)
Just dump stuff somewhere where it will break down through the healthy soil, and benefit an intentionally planted plant. Bit of randomness is good.

Mimic the healthiest and most effective natural rhythms.
Sometimes this means dumping more than one thing in one spot ...
But the modern compost crafting is, well, a vast improvement on chemical gardening for sure. And it gives you some sweet black gold.
... But, it is also craft, not nature. That implies work.

I have plenty of work to do. Do I have to also turn the compost? Or can I set it up to just "do its thing" without me?
Lazy is much my preference.

I make my heaps in a natural semi sphere I layer some and some I make a heap of browns and then dig out the center and place Nitrogen rich components in the hole, cover with the removed browns and water the heap till it leaks water at the base.
I don't turn heaps, I add air channels with a 3/4 inch ID pipe, just shove it in till it hits soil and remove. I go all the way around the heap when doing this. Compost is finished in 6 months or less, depending on how much Nitrogen I have added to the heap.
I also make square heaps which are the ones I compost bones and meat scraps in. These are layered as they are built but I still do the hole down the center (I use the pipe to wallow out the hole in these heaps) I add in spent coffee grounds and greens, put a cap on of browns and water.
The square heaps will need a turn every 3 weeks, I also cover these with a piece of old carpet. The squares are made inside three pallets which are tied at the corners with rope so I can easily take the container apart for turning or scooping out for use of the finished compost. This method takes around 6 months also.

I used to use a wood tumbler method, it was good but took money to build and was not as easy peasy as my current methods.

The geometry of a compost pile is very important.

There are many possible shapes, depending on where you are and your climate.

In most temperate climates, an egg-shaped ("igloo") or pyramidal (triangular) shape would work well. If it is an dry climate, you might want to cut the tip off the pyramid and make an indentation for water to collect. If it is too dry, you might have to use a compost pit to maintain good moisture. The fastest way to kill a pile is to let it dry out.

Over 120F proteins (enzymes) start to break down. Yes, it means you can kill seeds, but you can kill a lot of other good things too. But as long as it stays below 140F and above 108F, you will get some great activity.

If you are in the tropics, you probably wouldn't want to compost exclusively above ground at all. In the tropics, the constant outer warmth causes exceedingly fast decomposition -- part of why there's so little organic matter in rainforest soil. You might actually want to make a compost "pit" in the tropics, like Sir Albert Howard did during the dry seasons. During the monsoons he'd move it all to short hills under a covering.

In winter, you will probably want to stack taller to generate more warmth and favor a pointed top.
In summer, you will probably want to add more water; you might want a rounded top or a trench if you have dry summers. If you have wet summers, you might still want a domed top so that the tip doesn't overheat.

The best times for a compost pile to be build will be spring and fall. In the fall, you will have a naturally "colder" pile because many of the weeds and grasses will have flowered and started to go to seed: a much higher C/N ratio, which is not a bad thing. In the spring, the fresh green growth with have a lot more N. If you build a pile mid-Summer or mid-Winter, you will have opposite problems: in the summer, it is likely to get too hot and you will lose a lot of good organic matter. In the Winter, it is more likely to stay too cold/wet and get anaerobic which will result in leaching of valuable nitrogen. In the summer, you might add more carbonaceous materials to keep it cool; in the winter, you might add more manure to keep it warm. In the summer, you will almost certainly want to keep the pile in the shade; in the winter, the pile would almost certainly be benefited being situated on the south-facing side of a building or even in a greenhouse.

As a rule, every living organism has an inside and an outside and a skin to separate and mediate between those two realms. As a compost pile is built, it should always be covered with a layer of good topsoil. If the soil is sandy, the layer should probably be thicker. If the soil is clayey, the layer will not need to be so thick. As an extra measure, a breathable "fleece" should be used to cover the pile to maintain its humidity (if dry outside) or to keep it dry enough (if very wet outside). I use old landscaping fabric, which is cheap if you get it secondhand.

If you have a sandy soil, the soil beneath the pile should probably be layered not only with dry woody material but with a good high-CEC (Cation Exchange Capacity) clay. Sandy soil will not have to worry nearly as much as clayey soils about air, the most difficult part of composting.
If you have a clayey soil, you might even add sand to help prevent the pile from getting "wet feet."

A compost pile should be about 10% topsoil by volume, possibly more if sandy.

Another thought for building a compost pile, assuming it is all at one time, is to put hotter/dryer elements (like rabbit manure) at the bottom/center and to put colder/wetter elements (like cow manure) towards the top. With airier elements at the bottom and sloppier wetter elements at the top, better air circulation can be encouraged.

Consider growing (inoculated) legumes like clover or Austrian Winter Peas around the compost site to encourage good N-fixing bacteria. Also consider growing Birch, Alder, or Black Locust near your compost pile to shade (moderate) the temperature & humidity but also to inject your pile with N-fixing nodules.