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1. The familiar trigonometric functions can be geometrically derived from a circle.

But what if, instead of the circle, we used a regular polygon?

In this animation, we see what the “polygonal sine” looks like for the square and the hexagon. The polygon is such that the inscribed circle has radius 1.

We’ll keep using the angle from the x-axis as the function’s input, instead of the distance along the shape’s boundary. (These are only the same value in the case of a unit circle!) This is why the square does not trace a straight diagonal line, as you might expect, but a segment of the tangent function. In other words, the speed of the dot around the polygon is not constant anymore, but the angle the dot makes changes at a constant rate.

Since these polygons are not perfectly symmetrical like the circle, the function will depend on the orientation of the polygon.

More on this subject and derivations of the functions can be found in this other post

Now you can also listen to what these waves sound like.

This technique is general for any polar curve. Here’s a heart’s sine function, for instance

1. Real men pray.

1. Rules to always being a gentleman.

1. Hands with Holes
• In Exodus 33, during the days of an old covenant, Moses asked to see the face of God. God told Moses that no man could see His face and live. Then God placed His hand over the face of Moses and passed by in all His glory. Let's fast forward now to the new covenant. What do the hands of God look like now? His hands bear the scars and the holes of His sacrifice for us. What if we dared to ask what Moses asked, to see the face of God. Could it be that He places His hands over our face and through those holes in His hands we can see the face of God. Hebrews 10: 19-20 says "We have confidence to enter the Most Holy Place by the blood of Jesus, by a new and living way opened for us through the curtain, that is, his body." The nails were driven through the hands of Jesus and that One Man died, so that all men could see the face of God.
1. Apostolic Christianity Pt. 3

DISCIPLESHIP

It’s been a few days since my last post on this topic, the weekend got crazy. But now, by the grace of God, I have the time, wifi, and patience to write another one!

Alan Hirsch (the guy pictured above) is an incredibly wise man in the area of discipleship. He is part of Verge Network, along with other popular authors/pastors like Francis Chan and David Platt. (I strongly encourage you all to check out their website)

Anyways, Alan Hirsch has countless video blurbs of discipleship and its relation to the church today. He stresses two points:

1. The difference between a church of disciples and a church of consumers
2. The importance of the Five-Fold Church in relation to strong, missional churches

Consumerism vs. Discipleship

The church is backwards. It’s full of consumers rather than disciples.

“You cannot build a church on consumers. They will desert you at a moment’s notice because they have no commitments beyond their own needs.” -Alan Hirsch

How true is that?

So many churchgoers choose their church based on the style and volume of worship, the pastor’s haircut, the demographic of the congregation, the location, how wealthy/poor it is, etc… And as soon as something changes, they are displeased and go somewhere else.

The Church, just like Christ, should be a commitment.

And the Church, just like Christ, should build disciples.

“If we don’t disciple, the culture sure will.” -Alan Hirsch

Our culture thrives on consumerism, it’s become the “alternative religion of our day.” It  has taught us to become selfish, shallow, and lazy. Why work hard for something genuine when you can easily buy a superficial substitution?

WE ARE CALLED TO BE SEPARATE FROM THE WORLD!

We should strive for more than what the earth settles for.

Jesus was able to change the world through His twelve disciples plus the other seventy-two. The church would not exist if they were consumers.

We are called to “go out and make disciples of all nations” and to “teach them to obey everything [Christ] has commanded [us].” -Matthew 28:19-20

To the Hebrews, worship meant obedience. And obedience meant becoming like Christ through discipleship.

The early church had three stages of discipleship.

1. Becoming a disciple. When first called to the faith, new believers would go through a season called catechesis. It is an intense time of teaching and training in the content of faith and, as you can imagine, a huge commitment.
2. Being discipled by others. After building knowledge of the faith, disciples would then be committed into an intentional community and be discipled by a discipler.
3. Making disciples. They then took on the role of disciplers and mentored, encouraged, and taught new believers who were learning to follow Christ.

The early church knew how to do it.

Apostolic christianity stresses the importance of discipleship. And without disciples, the five-fold ministry would fall apart.

The Five-Fold Church

“It was He who gave some to be apostles, some to be prophets, some to be evangelists, and some to be pastors and teachers, to prepare God’s people for works of service, so that the body of Christ may be built up until we all reach unity in the faith and in the knowledge of the Son of God and become mature, attaining to the whole measure of the fullness of Christ.” (vv. 11-13)

And that is the Five-Fold Church. Instead of one pastor leading an entire congregation, there is one apostle, one prophet, one evangelist, one pastor, and one teacher leading a congregation, each as according to their giftings by the Holy Spirit.

The Holy Spirit appoints these gifts to build up disciples and to raise up the church.

God designed the church to be a movement.

We are the church and Christ is the Five-Fold. He is the greatest apostle, prophet, evangelist, pastor, and teacher to ever walk this earth. So why not base our ministries after His?

Discipleship is not something that should be taken lightly. Christ said:

“‘If anyone comes to me and does not hate his father and mother, his wife and children, his brothers and sisters-yes even his own life-he cannot be my disciple. And anyone who does not carry his cross and follow me cannot be my disciple.” -Luke 13:26-27

Becoming a true disciple rather than a church goer is a huge commitment, and Jesus encourages us to count the cost before fully following Him. If we are not willing to give up everything we have, we cannot be a disciple.

“The harvest is plentiful, but the workers are few. Ask the Lord of the harvest, therefore, to send out workers into His harvest field. Go! I am sending you out like lambs among wolves.”-Luke 10:2-4

2. (via imgTumble)

2. Gos is always on time.

1.  Camera Canon EOS 5D Mark II ISO 100 Aperture f/3.2 Exposure 1/100th Focal Length 50mm

Jesus Christ loves you.