# The Five Cardinals

I've got five cardinals, one without its mate.

My house is made of exactly four-hundred and thirty-five nice solid bricks.

(I'm kidding - they're rather fluffy.)

My length is about two hundred and forty-three, give or take.

My bench has nine legs - really, no tricks!

Mathematically, I have some interesting attributes, too:

a hyperbolic trig function in my middle, a quadratic on my left,

and - once a year - a sphere on my right. Yoo-hoo!

My whole is, in a way, a complicated plane shape who from 2D has left.

Note: just for fun, I made a question with the knowledge and trivia tags - see here for why.

Note: I replaced "chair" with "bench". I think it's a little more accurate this way.

Hint 1:

The specific trig function is the hyperbolic cosine function. It has a very famous application. That application is what that line's clue is driving at.

Hint 2:

The sphere exists every 365 days (366 every four years, of course), and is used to celebrate a certain event that happens with that regularity. It is destroyed at the end (or beginning?) of the event.

Edit - Update

I think the answer is

The United States of America

I've got five cardinals, one without its mate.

Five of the U.S. states have cardinal directions in their name: North Carolina, South Carolina, North Dakota, South Dakota and West Virginia - which is the one without the mate as there is no East Virginia.

Original: Five major self-governing territories - two are located in Micronesia and two in the Caribbean. American Samoa, in Polynesia, does not have a mate.

My house is made of exactly four-hundred and thirty-five nice solid bricks.

The U.S. House of Representatives has 435 voting representatives.

(I'm kidding - they're rather fluffy.)

My length is about two hundred and forty-three, give or take.

This year it is 243 years since 1776, when the United States declared independence

My bench has nine legs - really, no tricks!

The Supreme Court of the U.S. has nine justices on its bench.

Mathematically, I have some interesting attributes, too:

a hyperbolic trig function in my middle,

cosh (as confirmed by OP). I think this refers to the catenary curve, a famous example of which is the Gateway Arch in St. Louis, Missouri.

a quadratic on my left,

A famous example of a parabolic arch is the Golden Gate Bridge in California.

and - once year - a sphere on my right. Yoo-hoo!

This is the Times Square Ball in New York, which drops to celebrate the New Year

My whole is, in a way, a complicated plane shape who from 2D has left.

Geographically, coastlines are often referred to as fractal in nature so may be attributed with a fractal dimension.
Alternatively, because the U.S. is quite a large country, the amount it curves over the surface of the Earth is quite large compared to most other countries so we can say it is definitely not flat (2D).