I used the JsFiddle at jsfiddle.net/5tzk3/10. I changed it to display the div as square shaped dialog (both horizontally and vertically centered). The result is at jsfiddle.net/5tzk3/548.
As you see, centering horizontally was easy, but I could not get it centered vertically. Anyone who knows how to do that with pure CSS?
Edited code below:
<div class="blind">
    <div class="wrapper">
        <div class="main">
            I'm your div with an aspect-ratio of 1:1!
        </div>
    </div>
</div>
html, body, .blind {
    height: 100%;
    width: 100%;    
}
.blind {
    left: 0;
    position: fixed;
    text-align: center;
    top: 0;
}
.wrapper {
    display: inline-block;
    margin: auto;
    position: relative;
    width: 50%;
}
.wrapper:after {
    content: '';
    display: block;
    padding-top: 100%;
}
.main {
    background-color: rgb(0, 162, 232);
    bottom: 0;
    left: 0;
    margin: auto;
    position: absolute;
    right: 0;
    top: 0; 
}
 
     
     
     
    