いく
iku
In Japanese you 'go' to the person calling you: いく
The picture Moving away from the spot where you are standing right now. いく moves away from here; くる moves from over there toward here.
Where the picture shows up in いく
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you head over there
明日、病院に行きます。
ashita, byōin ni ikimasu
I'm going to the hospital tomorrow.
Stepping out your front door and receding toward the hospital, back turned
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you move toward the other person
今すぐそちらに行きます。
ima sugu sochira ni ikimasu
I'll be right there.
Leaving the spot you're standing on and moving your feet toward where they are
English says 'I'm coming', Japanese says 行きます. The reference point is always where you are now.
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the work rolls along
新しい仕事はうまくいっていますか。
atarashii shigoto wa umaku itte imasu ka
Is the new job going well?
A cart you gave one push rolling on ahead by itself
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you take it along
傘を持っていきます。
kasa o motte ikimasu
I'll take an umbrella with me.
The hand holding the umbrella moving off into the distance along with you
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it keeps going that way
これからもっと寒くなっていきます。
kore kara motto samuku natte ikimasu
It'll keep getting colder from now on.
The temperature mark sliding further down and away from where it stands now
All of it in one picture
いく is the picture of moving away from the spot where you stand right now. You recede toward the hospital with your back turned; you leave this spot and step toward the other person; the work rolls on ahead like a cart you pushed once; the umbrella travels off with you in your hand; the cold keeps sliding further down and away from today. Only what recedes changes — your body, a job, a thing you carry, a trend. So 〜ていく keeps this moment as the anchor and points away from it, forward.