CSS flexbox vertically/horizontally center image WITHOUT explicitely defining parent height

Just add the following rules to the parent element:

display: flex;
justify-content: center; /* align horizontal */
align-items: center; /* align vertical */

Here's a sample demo (Resize window to see the image align)

Browser support for Flexbox nowadays is quite good.

For cross-browser compatibility for display: flex and align-items, you can add the older flexbox syntax as well:

display: -webkit-box;
display: -webkit-flex;
display: -moz-box;
display: -ms-flexbox;
display: flex;
-webkit-flex-align: center;
-ms-flex-align: center;
-webkit-align-items: center;
align-items: center;

Without explicitly defining the height I determined I need to apply the flex value to the parent and grandparent div elements...

<div style="display: flex;">
<div style="display: flex;">
 <img alt="No, he'll be an engineer." src="theknack.png" style="margin: auto;" />
</div>
</div>

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If you're using a single element (e.g. dead-centered text in a single flex element) use the following:

align-items: center;
display: flex;
justify-content: center;