This spelling rule applies only to words where the two letters form a diphthong, which is the sound of two vowels combined in a single syllable, like the AE in Caesar. The word science does not contain a diphthong; the I and the E are separate sounds in separate syllables. Nor does society contain a diphthong, nor do deity or quiet. We can, however, see the diphthong in hieroglyphics.
The I before E rule does not apply to plurals like vacancies, nor does it apply to comparatives and superlatives like jucier and juiciest. Which leaves fewer than a dozen anomalies, such as ancient and glacier. Even these few exceptions are subject to argument, however, depending on how the words are pronounced.
Here’s the rest of the I before E rhyme:
I before E, except after C.
Or when sounding like ay in neighbor and weigh
Or when sounding like eye in seismic and height.
Read that little rhyme over again, and you will clearly see that the rule applies only to words that have the ee sound. It does not apply to words with the ay sound: eight, deign, vein, sleigh. Nor does it apply to words that have the eye sound: feisty, eidetic, sleight, einsteinium.
The problem comes when we have words with different sounds: sieve, friend, view, patient, efficient. These words usually follow the I before E rule.
The rule is also a guide to pronunciation. You may pronounce either any way you wish (ee-ther or eye-ther), but notice that it does fall into the eye group of words like eiderdown and Eiffel Tower. As for geisha and sheik, they both have the ay sound.